tag:blogger.com,1999:blog-62500751253152313372024-02-19T05:22:02.903-08:00Sumatoria y Productoria_Natalia_http://www.blogger.com/profile/01817418121318373428noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6250075125315231337.post-50935905193859534102009-05-04T12:00:00.000-07:002009-05-18T12:38:32.166-07:00_Propiedades de la Productoria_<div style="text-align: left; color: rgb(102, 0, 204);"><ul><li><span style="font-style: italic; font-weight: bold;"> Propieades Multiplicativas:</span></li></ul><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJnjJOc6_v-nwg-E3LEgQ90olDj1xHTExu53Cp3qY0kwpG7AgeedmOZL0BjzX27C2sW3OKiQLJ6vgsOyMBRVfqiZMMNLClhogB8k82q7O0kECATD6leeLfC7Rd_Xx1kK-XQ2jSylJrFJFu/s1600-h/pro1.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 229px; height: 49px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJnjJOc6_v-nwg-E3LEgQ90olDj1xHTExu53Cp3qY0kwpG7AgeedmOZL0BjzX27C2sW3OKiQLJ6vgsOyMBRVfqiZMMNLClhogB8k82q7O0kECATD6leeLfC7Rd_Xx1kK-XQ2jSylJrFJFu/s320/pro1.png" alt="" id="BLOGGER_PHOTO_ID_5332048441787471474" border="0" /></a><br /><br /><br /><br /><br /><span style="font-weight: bold;"><span style="font-weight: bold;"><span style="color: rgb(0, 0, 0);">Demostarción por inducción:<br /></span></span></span><br /><br /><span style="color: rgb(0, 0, 0);">* Tomemos </span><span style="font-weight: bold;"><span style="font-weight: bold;"><span style="color: rgb(0, 0, 0);">n=1 </span></span></span><span style="color: rgb(0, 0, 0);">y veamos si se cumple la Siguien</span><span style="color: rgb(0, 0, 0);">t</span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUDyTuXgKs15qcTI7-nsuSPTQpiposS31plYuVxJxzxYLAVJ2NOj9bxmLElPtR1ufVj3S46ymknkUXV2jX36pUP3pgudyQWtU4CCT9SD_tSZNNmQo_O0zGqQ2uuM9by3LUDoqcwxQlshL2/s1600-h/pro6.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 231px; height: 51px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUDyTuXgKs15qcTI7-nsuSPTQpiposS31plYuVxJxzxYLAVJ2NOj9bxmLElPtR1ufVj3S46ymknkUXV2jX36pUP3pgudyQWtU4CCT9SD_tSZNNmQo_O0zGqQ2uuM9by3LUDoqcwxQlshL2/s320/pro6.png" alt="" id="BLOGGER_PHOTO_ID_5332057271863999938" border="0" /></a><span style="color: rgb(0, 0, 0);">e igualdad </span><br /><br /><br /><br /><br /><span style="color: rgb(0, 0, 0);"><br /></span><br /><span style="color: rgb(0, 0, 0);">y la igualdad es cierta para <span style="font-weight: bold;">n=1</span></span><div style="text-align: center; color: rgb(51, 51, 255);"><div style="text-align: center;"><span style="color: rgb(51, 153, 153); font-weight: bold;"> " La <span style="font-style: italic;">inducción</span> es un razonami</span><span style="color: rgb(51, 153, 153); font-weight: bold;">ento que permite demostrar una infinidad de proposiciones,dependiendo de un para</span><span style="color: rgb(51, 153, 153); font-weight: bold;">metreo </span><span style="font-style: italic; color: rgb(51, 153, 153); font-weight: bold;">n</span><span style="color: rgb(51, 153, 153); font-weight: bold;"> que toma una infinidad de valores </span><span style="color: rgb(51, 153, 153); font-weight: bold;">enteros" </span><br /></div><div style="text-align: left;"><br /><span style="color: rgb(0, 0, 0);">* Supongásmola sierta para <span style="font-weight: bold;">n</span> y analicém</span><span style="color: rgb(0, 0, 0);">osla para</span><span style="color: rgb(0, 0, 0);"> <span style="font-weight: bold;">n+1</span></span><span style="font-weight: bold;"><br /></span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsFcNlQNixBk3PXuBuT_Dezn74oCLpyZq7gKCXRStblPqBc0vdK3BHIJ3QYHutyMy6rYn_32rTHZgTpTDQ5eVxwyLWsCr7zpCypStiyiUqAQWaDdOJASEycTGQyhnhxbjSiTik2qIdU0gL/s1600-h/pro5.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 320px; height: 46px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsFcNlQNixBk3PXuBuT_Dezn74oCLpyZq7gKCXRStblPqBc0vdK3BHIJ3QYHutyMy6rYn_32rTHZgTpTDQ5eVxwyLWsCr7zpCypStiyiUqAQWaDdOJASEycTGQyhnhxbjSiTik2qIdU0gL/s320/pro5.png" alt="" id="BLOGGER_PHOTO_ID_5332055066353125602" border="0" /></a></div></div><br /><br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDCG8quyFVY-nDqUWOcngVACiy8GFz3tCqrDXWm2XOwM4ErGGIpOf0TD59kX8OGQsk-REgvCSIkZlP70mSRykPKT1tTFtz5fuGLNPwC360z0HKaQWYUPWPt0Yy1NHqqCvT0y_w6zDh1kSQ/s1600-h/pro4.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 303px; height: 51px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDCG8quyFVY-nDqUWOcngVACiy8GFz3tCqrDXWm2XOwM4ErGGIpOf0TD59kX8OGQsk-REgvCSIkZlP70mSRykPKT1tTFtz5fuGLNPwC360z0HKaQWYUPWPt0Yy1NHqqCvT0y_w6zDh1kSQ/s320/pro4.png" alt="" id="BLOGGER_PHOTO_ID_5332054234962554114" border="0" /></a><br /><br /><br /><br /><br /><br /><span style="font-weight: bold; color: rgb(0, 0, 0);">( Definición por Inducción)</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWXnhwF1lH9jBXM8-Bl6Lzfug88v2Ff-heoaoDeEGFuD1aSn822n3Ib583L6JiJO86yr7Ki_8f5_UfEnatmIUrydWqZToz-pumqwSntW3c18EhoRPQZzxHWIWQgTYQib1USSWySndLMPH6/s1600-h/pro3.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 285px; height: 51px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWXnhwF1lH9jBXM8-Bl6Lzfug88v2Ff-heoaoDeEGFuD1aSn822n3Ib583L6JiJO86yr7Ki_8f5_UfEnatmIUrydWqZToz-pumqwSntW3c18EhoRPQZzxHWIWQgTYQib1USSWySndLMPH6/s320/pro3.png" alt="" id="BLOGGER_PHOTO_ID_5332053377014218578" border="0" /></a><br /><br /><br /><br /><span style="font-weight: bold; color: rgb(0, 0, 0);">(Asociatividad de IR)</span><br /><br /><br /><span style="color: rgb(0, 0, 0);">Luego,</span><br /><span style="color: rgb(0, 0, 0);"><br /></span><div style="text-align: left;"><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3UTL6h7olmLBDt2ScD8jEBl1HEZD741rRwlLZpbhtDAWYf3RuLRS02BAv-99CEbFVu4Smg6ac2NyV8sREYsxd-_lmx8OqIWjBll-U52H7DzAT_NGyA6uJEYaeRueG5T9gqqoXwyx1yVUT/s1600-h/pro2.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 289px; height: 52px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3UTL6h7olmLBDt2ScD8jEBl1HEZD741rRwlLZpbhtDAWYf3RuLRS02BAv-99CEbFVu4Smg6ac2NyV8sREYsxd-_lmx8OqIWjBll-U52H7DzAT_NGyA6uJEYaeRueG5T9gqqoXwyx1yVUT/s320/pro2.png" alt="" id="BLOGGER_PHOTO_ID_5332051474889127298" border="0" /></a></div><br /><span style="font-weight: bold;"><br /><br /></span><br /><br /><br /><ul style="font-style: italic;"><li><span style="font-weight: bold;"> Propiedad Telescópica:</span></li></ul><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh-atS5Tl10HvTpcOwPXhTzTNEASksIWnNgFkYrWbFXSc_XGMd0_JLESIoJRFUFa2JVT3o4a264QwA7fzX_53ub-PM-P0mskrPcbMTR1s2lLK7b0ToMTsdBy0WI9POUfVsFl_4qqmSPKL-/s1600-h/pro1.png"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 118px; height: 49px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh-atS5Tl10HvTpcOwPXhTzTNEASksIWnNgFkYrWbFXSc_XGMd0_JLESIoJRFUFa2JVT3o4a264QwA7fzX_53ub-PM-P0mskrPcbMTR1s2lLK7b0ToMTsdBy0WI9POUfVsFl_4qqmSPKL-/s320/pro1.png" alt="" id="BLOGGER_PHOTO_ID_5332058882920014930" border="0" /></a><br /><span><span style="color: rgb(0, 0, 0);">si cada</span></span><span style="font-weight: bold;"><span style="color: rgb(0, 0, 0);"> </span></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2M3oio59XPqjb3uruy2YvOirLuWuAPmCNf1Z68lD4N2kwoqrg3bgyhKefw3Y3EEBoppDAj_WB7FSmV0HVJyhPiYZugj1vN7s6HFNjOL12St8ivD79TlnvRG0a2GHBEFog0DBn98wWeBZl/s1600-h/pr.png"><img style="cursor: pointer; width: 54px; height: 20px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2M3oio59XPqjb3uruy2YvOirLuWuAPmCNf1Z68lD4N2kwoqrg3bgyhKefw3Y3EEBoppDAj_WB7FSmV0HVJyhPiYZugj1vN7s6HFNjOL12St8ivD79TlnvRG0a2GHBEFog0DBn98wWeBZl/s320/pr.png" alt="" id="BLOGGER_PHOTO_ID_5332059335741654162" border="0" /></a><br /><br /><br /><span style="font-weight: bold; color: rgb(0, 0, 0);">Demostración por Inducción</span><br /><br /><span style="color: rgb(0, 0, 0);">Analicemoslo para <span style="font-weight: bold;">n=1</span></span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0EIyukDv4o2xl37O_WMCQiz2tvipqcmN8dDCU9kzYdFV1gTBunRGwNZNmqiNxk3hb6Da-Hed3SVd7GNWWFi8CoVVqujJC7n8-DTX6mDPCT6vokCghfOmRVBBE7nikTJcBLkmuZVgAV2dZ/s1600-h/PRO.png"><img style="cursor: pointer; width: 118px; height: 52px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0EIyukDv4o2xl37O_WMCQiz2tvipqcmN8dDCU9kzYdFV1gTBunRGwNZNmqiNxk3hb6Da-Hed3SVd7GNWWFi8CoVVqujJC7n8-DTX6mDPCT6vokCghfOmRVBBE7nikTJcBLkmuZVgAV2dZ/s320/PRO.png" alt="" id="BLOGGER_PHOTO_ID_5337243836627363938" border="0" /></a><span style="font-weight: bold; color: rgb(0, 0, 0);"> Con</span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1Ir8pvEPpzCiB5DziwKT2mcT70_WLjPyFIRTf1izO-NPTVH_y53POOPo9uD063Ac_woiQN8qK2ms0EayDMiLmdeZEbY6Um0mee0teq7gFl4PQT646rtyfXIP1tsCrHgnC-zp3-cg5MnO8/s1600-h/25.png"><img style="cursor: pointer; width: 52px; height: 19px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1Ir8pvEPpzCiB5DziwKT2mcT70_WLjPyFIRTf1izO-NPTVH_y53POOPo9uD063Ac_woiQN8qK2ms0EayDMiLmdeZEbY6Um0mee0teq7gFl4PQT646rtyfXIP1tsCrHgnC-zp3-cg5MnO8/s320/25.png" alt="" id="BLOGGER_PHOTO_ID_5337244338819200274" border="0" /></a> <span style="color: rgb(0, 0, 0);">y la igualdad es sierta para n=1</span><br /><br /><span style="color: rgb(0, 0, 0);">* Supong</span><span style="color: rgb(0, 0, 0);">ámosla cierta para n y </span><span style="color: rgb(0, 0, 0);">analicémosla para n+1</span><br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhT9ZCdo2JBoEQTB7YV4n-kGVaB78wpDbofIwet8jB1q8_5GytO-xsVI5YT8RSKs3q0aGqHk3PrtKmcOk0ueMJ4QuRjrfzv0Uu0GMCaSA2CFWQEwRU_PWT5b3HKapHmaWct19m2dQ9Hd_Su/s1600-h/26.png"><img style="cursor: pointer; width: 237px; height: 52px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhT9ZCdo2JBoEQTB7YV4n-kGVaB78wpDbofIwet8jB1q8_5GytO-xsVI5YT8RSKs3q0aGqHk3PrtKmcOk0ueMJ4QuRjrfzv0Uu0GMCaSA2CFWQEwRU_PWT5b3HKapHmaWct19m2dQ9Hd_Su/s320/26.png" alt="" id="BLOGGER_PHOTO_ID_5337247012478113986" border="0" /></a><span style="color: rgb(0, 0, 0);"> ( definición por inducción)</span><br /><span style="color: rgb(0, 0, 0);"><br /><br />Luego,</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3TGCSUXDp105joG3hCCwnhnP1O5IEFu3kHfWAXAmlBFsfL7VcREkPsjJfUKTTD5OaIXHMZjTrNYbDqlue2ASZ6p4znHIE6PbueinVkuTrN1CbGY0olV9RdES-WUi8eKvVxP5W82MZiC2X/s1600-h/27.png"><img style="cursor: pointer; width: 165px; height: 53px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3TGCSUXDp105joG3hCCwnhnP1O5IEFu3kHfWAXAmlBFsfL7VcREkPsjJfUKTTD5OaIXHMZjTrNYbDqlue2ASZ6p4znHIE6PbueinVkuTrN1CbGY0olV9RdES-WUi8eKvVxP5W82MZiC2X/s320/27.png" alt="" id="BLOGGER_PHOTO_ID_5337247540125363170" border="0" /></a><span style="color: rgb(0, 0, 0);"> que es lo queríamos demostrar.</span><br /><br /><br /><span style="color: rgb(0, 0, 0);"><br />Nótese que nuestra exigencia era que para cada k</span>,<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJjCaVjrqNLY11nUMeWKiSbsZEWBoC1xVCHie0secpPk8AJCsxZZe3D-vkppxTdIfk9pYYXhiHt0bnWQtuemDv_48E_abl99PBqS8shUQ35o4hpNwq8cX0wFjGsx0lL2yiVZ8qXqT-BBF8/s1600-h/28.png"><img style="cursor: pointer; width: 54px; height: 20px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJjCaVjrqNLY11nUMeWKiSbsZEWBoC1xVCHie0secpPk8AJCsxZZe3D-vkppxTdIfk9pYYXhiHt0bnWQtuemDv_48E_abl99PBqS8shUQ35o4hpNwq8cX0wFjGsx0lL2yiVZ8qXqT-BBF8/s320/28.png" alt="" id="BLOGGER_PHOTO_ID_5337248626371685026" border="0" /></a> .<br /><span style="color: rgb(0, 0, 0);">En particular para</span> <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaSEwSgcrV5Xsx23_KPpCnt02XNr6g5xf4L6ZJJq3_J18ICIzpjEBfY3vv3lSWsQMlSAw5sBTGQ11oqaO8kwJYRpObKjSprdVFgvG12R8eka03dsp7p2cddkcWA1CxZmJ7TsC46mLyh8tv/s1600-h/d9.png"><img style="cursor: pointer; width: 48px; height: 14px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaSEwSgcrV5Xsx23_KPpCnt02XNr6g5xf4L6ZJJq3_J18ICIzpjEBfY3vv3lSWsQMlSAw5sBTGQ11oqaO8kwJYRpObKjSprdVFgvG12R8eka03dsp7p2cddkcWA1CxZmJ7TsC46mLyh8tv/s320/d9.png" alt="" id="BLOGGER_PHOTO_ID_5337249400373986322" border="0" /></a> ; <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizg3b1Lgy0XrBMVaMJup6PrvOiB8Emq0W9YiVQx9suB7-d6rW_YH2jTAn1CpPICC6JYExDDUFYMFSi4_MlalWWMM6Xw3s7BEnH3SuhCcUC-G3qqtz4YuP1Do5drjzS2R2zMtwMeOIJRaSo/s1600-h/29.png"><img style="cursor: pointer; width: 100px; height: 20px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizg3b1Lgy0XrBMVaMJup6PrvOiB8Emq0W9YiVQx9suB7-d6rW_YH2jTAn1CpPICC6JYExDDUFYMFSi4_MlalWWMM6Xw3s7BEnH3SuhCcUC-G3qqtz4YuP1Do5drjzS2R2zMtwMeOIJRaSo/s320/29.png" alt="" id="BLOGGER_PHOTO_ID_5337249066528647586" border="0" /></a>. <span style="color: rgb(0, 0, 0);">Luego la Simplificación es posible y</span><br /><br /><br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIlAbFC1SjRzNJ8AROlNKKyrNnyu_KBuJMFFSypRIFs5A046ehrrntb85U_6Xws_ujBXTZBrb-D7IrPJVIjKwvkQOQ-TECwUO_PTJM6HE3qwvdujCZLqTqSgoXsh8oD3ogd9sEpX8GQZP-/s1600-h/30.png"><img style="cursor: pointer; width: 140px; height: 53px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIlAbFC1SjRzNJ8AROlNKKyrNnyu_KBuJMFFSypRIFs5A046ehrrntb85U_6Xws_ujBXTZBrb-D7IrPJVIjKwvkQOQ-TECwUO_PTJM6HE3qwvdujCZLqTqSgoXsh8oD3ogd9sEpX8GQZP-/s320/30.png" alt="" id="BLOGGER_PHOTO_ID_5337250199940635858" border="0" /></a><br /><br /><br /><br /><br /><br /><br /><br /></div>_Natalia_http://www.blogger.com/profile/01817418121318373428noreply@blogger.com1tag:blogger.com,1999:blog-6250075125315231337.post-58974733127899462009-04-27T12:33:00.000-07:002009-04-27T12:52:51.466-07:00_Productoria_Tambíen conocida como <span style="color: rgb(51, 0, 153); font-style: italic;">Multiplicatoria</span> o <span style="color: rgb(51, 0, 153); font-style: italic;">Pitatoria</span>, este es un operador matematico el cual consiste en la multiplicación finita o infinita de factores mediante un simbolo matematico que simplifica la operación, llamado <span style="color: rgb(102, 0, 204); font-weight: bold;">Simbolo Productorio.</span><br /> Se puede definir como:<br /><dl><dd> <dl><dd><img class="tex" alt="\prod_{k=1}^1 a_k = a_1" src="http://upload.wikimedia.org/math/c/5/1/c513aefde83dbdc4af3dfb1494f69e9c.png" /></dd><dd><br /></dd><dd><br /></dd><dd>Supuesta definida para un <span style="font-weight: bold;">n ≥ 1</span> fijo, se define </dd></dl></dd></dl><dl><dd> <dl><dd><img class="tex" alt="\prod_{k=1}^{n+1} a_k = (\prod_{k=1}^n a_k)a_{n+1}" src="http://upload.wikimedia.org/math/c/1/e/c1ee27a1d70265832688f0753930d1f9.png" /></dd></dl></dd></dl><br /><br />Ejemplos:<br /><br /><br /> Se puede tomar <b>n=1</b> y aplicar la segunda igualdad para obtener:<br /><br /><img class="tex" alt=" \prod_{k=1}^2 a_k = (\prod_{k=1}^1 a_k)(a_2) = a_1a_2 " src="http://upload.wikimedia.org/math/b/4/8/b4837871a427308e11e81f51cc36fcbc.png" />.<span style="font-weight: bold;"><br /><br /><br /></span>Definida para <b>n=2</b>, se puede aplicar otra vez la segunda igualdad con <b>n=2</b> para luego obtener<br /><br /><br /><br /><img style="width: 281px; height: 54px;" class="tex" alt=" \prod_{k=1}^3 a_k = (\prod_{k=1}^2 a_k)(a_3) = (a_1a_2)a_3 " src="http://upload.wikimedia.org/math/3/4/0/34093110c6b55f93c467528af5e0e5d6.png" />.<br /><br /><br /><br />Por lo que el producto <img class="tex" alt="\mathit{(a_1a_2)a_3} \,\!" src="http://upload.wikimedia.org/math/d/5/4/d5434882d768788b1d5352f0c759ae97.png" /> es el mimso que <img class="tex" alt="\mathit{a_1(a_2a_3)} \,\!" src="http://upload.wikimedia.org/math/a/7/c/a7c7d5a4633ff04c85c042291314b8ad.png" /> y, por lo tanto, podemos prescindir del uso de paréntesis sin peligro de confusión y usar simplemente<br /><br /><img class="tex" alt="\mathit{a_1a_2a_3} \,\!" src="http://upload.wikimedia.org/math/c/f/a/cfa39be59185e66a4a6b3aa3f4e785e9.png" /> para <img class="tex" alt=" \prod_{k=1}^3 a_k " src="http://upload.wikimedia.org/math/3/c/e/3ceb80ec5145514a61b3951d698b574a.png" />.<br /><br /><br /> Entonces, usar este razonamiento para cualquier <img class="tex" alt=" n \in \mathbb{N} " src="http://upload.wikimedia.org/math/2/e/2/2e259ed38824a8713d454b6f5bad6bde.png" /> sin que haya peligro de confusión.<br />Luego, se puede aplicar la definición de Multiplicatoria, para definir <b>n!</b> <i>(n factorial)</i> como sigue:<br /><br /><p><img class="tex" alt=" \prod_{k=1}^n k = n! " src="http://upload.wikimedia.org/math/1/e/e/1eebebc61eef66862032c437dde5ace9.png" /></p><p>Se define <span style="color: rgb(102, 51, 102); font-weight: bold;">0!=1!=1</span><br /></p>_Natalia_http://www.blogger.com/profile/01817418121318373428noreply@blogger.com0tag:blogger.com,1999:blog-6250075125315231337.post-59253534467153253422009-04-20T12:07:00.000-07:002009-04-27T12:01:26.883-07:00_ Propiedades de la Sumatoria_<div><div><div><div><div><div><div><div><div><div><div><div><div><div><div><div align="center">Entre las propiedades generales de las sumatorias reportadas en la literatura se encuentra las once que se relacionan a continuación, cuya demostración se realiza utilizando el procedimiento matemático de Inducción Completa. </div><br /><br /><div style="font-weight: bold; color: rgb(51, 51, 255);" align="left">Procedimiento #1:</div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmHySets1Al7quExyIr-2s_VYR6JN2JFbExjA1dXIyBoqPkohvhEnVC_noIHYgii7DHIWhJ9hv_UiG6Y79thTutxT4rDSriVLxGNElu_QanQ4W9kodxDSV_RU5m58TOGOd5Nw1JIi_7tw2/s1600-h/pro1.gif"><img id="BLOGGER_PHOTO_ID_5326862158044801970" style="margin: 0px 10px 10px 0px; float: left; width: 99px; height: 45px;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmHySets1Al7quExyIr-2s_VYR6JN2JFbExjA1dXIyBoqPkohvhEnVC_noIHYgii7DHIWhJ9hv_UiG6Y79thTutxT4rDSriVLxGNElu_QanQ4W9kodxDSV_RU5m58TOGOd5Nw1JIi_7tw2/s320/pro1.gif" border="0" /></a><div><br /><br /><br /><br /><span style="font-weight: bold; color: rgb(51, 51, 255);">Procedimiento #2:</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXA_h2CVtLmvZh63x3dslgGdqmKE5FHxnUTCwom1t32G6tJ0EBgDY3-0anqkjY6fiSXpXHwXrNwXtVbJJngp3a6MyTF9CnkFYj31dWU8kH7pLttwJ60z-JSiqkSDV-UIAuPVNI0EhrEewV/s1600-h/pro2.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 157px; height: 48px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXA_h2CVtLmvZh63x3dslgGdqmKE5FHxnUTCwom1t32G6tJ0EBgDY3-0anqkjY6fiSXpXHwXrNwXtVbJJngp3a6MyTF9CnkFYj31dWU8kH7pLttwJ60z-JSiqkSDV-UIAuPVNI0EhrEewV/s320/pro2.gif" alt="" id="BLOGGER_PHOTO_ID_5329445848486449890" border="0" /></a><br /><br /><br /><br /><span style="font-weight: bold; color: rgb(51, 51, 255);">Procedimiento #3:</span></div></div><div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ_DY8TjOwaNOxdLyqzxtaWUonCtyLjRVw4zuFHuN5n35bqT-bB0XADXuZmR0COxejwfbB1hrT29rNTRpm0Y8tncRn7briA6sEM6Ub_kqwelKH8RRIeruKHfsfvGEIK8iixgMenu1tofDj/s1600-h/pro3.gif"><img id="BLOGGER_PHOTO_ID_5326862465661320050" style="margin: 0px 10px 10px 0px; float: left; width: 99px; height: 45px;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ_DY8TjOwaNOxdLyqzxtaWUonCtyLjRVw4zuFHuN5n35bqT-bB0XADXuZmR0COxejwfbB1hrT29rNTRpm0Y8tncRn7briA6sEM6Ub_kqwelKH8RRIeruKHfsfvGEIK8iixgMenu1tofDj/s320/pro3.gif" border="0" /></a><br /><br /><br /><br /><br /><span style="font-weight: bold; color: rgb(51, 51, 255);">Procedimiento #4:</span><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjINmNGNmDyMHHJa2RIpc9hRecIvZFiyznxchyphenhyphenz5Kfm4oDZW9EXpxwUBpVJDzO4PMGMFljOm98FHxsETGw-GKCXKnCSTq12VQ3XJayNyebm01A9YvAWOejpCqvr3csDp7z35RNtaxAXn6I9/s1600-h/pro4.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 96px; height: 45px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjINmNGNmDyMHHJa2RIpc9hRecIvZFiyznxchyphenhyphenz5Kfm4oDZW9EXpxwUBpVJDzO4PMGMFljOm98FHxsETGw-GKCXKnCSTq12VQ3XJayNyebm01A9YvAWOejpCqvr3csDp7z35RNtaxAXn6I9/s320/pro4.gif" alt="" id="BLOGGER_PHOTO_ID_5329446114634379746" border="0" /></a><br /><br /></div></div><p><br /></p><p><br /></p><p><span style="color: rgb(51, 51, 255); font-weight: bold;">Procedimiento #5</span><br /><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKdGl6rThJJ7_QCg90LXXeVNkzuvSYObfjFpkGGWEhBcdnv4xTZOAKNc9yFw4nCfmTHmXZqBsClGw4iTh8ayvZv8vdSVfoYsIx6en7UgBk1yrlGY-TALSwJQWDUaXD_Wn25B9VC3j00tQv/s1600-h/pro5.gif"><img id="BLOGGER_PHOTO_ID_5326864232273117138" style="margin: 0px 10px 10px 0px; float: left; width: 136px; height: 45px;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKdGl6rThJJ7_QCg90LXXeVNkzuvSYObfjFpkGGWEhBcdnv4xTZOAKNc9yFw4nCfmTHmXZqBsClGw4iTh8ayvZv8vdSVfoYsIx6en7UgBk1yrlGY-TALSwJQWDUaXD_Wn25B9VC3j00tQv/s320/pro5.gif" border="0" /></a></p></div></div><br /><br /><br /><br /><span style="font-weight: bold; color: rgb(51, 51, 255);">Procedimiento #6:</span><br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3fCgbJij4BTCMjV5w-8FueHXB2vBcLqdxUFFcySAO44_eeqBfN76_eG8TkAAG_3K3E2vlBdmI0DaEea6fMmYK3WsGXg9AWfg3NVRe04x-jLjd1hu7szztHro4Q_9pKirk5KRgmniTx51Z/s1600-h/pro6.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 108px; height: 45px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3fCgbJij4BTCMjV5w-8FueHXB2vBcLqdxUFFcySAO44_eeqBfN76_eG8TkAAG_3K3E2vlBdmI0DaEea6fMmYK3WsGXg9AWfg3NVRe04x-jLjd1hu7szztHro4Q_9pKirk5KRgmniTx51Z/s320/pro6.gif" alt="" id="BLOGGER_PHOTO_ID_5329446412952115426" border="0" /></a><br /><br /><br /><br /><p><span style="font-weight: bold; color: rgb(51, 51, 255);">Procedimiento #7:</span><br /></p></div><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYc1Q0COwptzzPUSQrFiDPVxw8bZcBxW4YHD2yt6fGD7TRzqGJNjWhKyRP8CdSjCKtF0mHaCKCoxEZ-h9FnH4D_4aYnYHouSVSap0Y9R9DY7VD7ojrom4RLfqYPXVFOfFrFSqyKlj51U0E/s1600-h/pro7.gif"><img id="BLOGGER_PHOTO_ID_5326863115666901042" style="margin: 0px 10px 10px 0px; float: left; width: 156px; height: 45px;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYc1Q0COwptzzPUSQrFiDPVxw8bZcBxW4YHD2yt6fGD7TRzqGJNjWhKyRP8CdSjCKtF0mHaCKCoxEZ-h9FnH4D_4aYnYHouSVSap0Y9R9DY7VD7ojrom4RLfqYPXVFOfFrFSqyKlj51U0E/s320/pro7.gif" border="0" /></a><span style="font-weight: bold;"><br /><br /></span><br /><span style="font-weight: bold;"><br /></span><span style="font-weight: bold; color: rgb(51, 51, 255);">Procedimiento #8:</span><span style="font-weight: bold;"><br /></span><br /><p><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_5DsI9IOebRhCLEwNL_AhDLHODoB8MbE0N3APON0uw1l_1ymJpJJlC7wEShQNw9UWWg1BWNFjUDsLZ74uxy7k4ung5hA0tRKjDCYoTtKIADP2lZU_gYdSZXkuVBul-SqFl4jZEFRrL6Jm/s1600-h/pro8.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 183px; height: 45px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_5DsI9IOebRhCLEwNL_AhDLHODoB8MbE0N3APON0uw1l_1ymJpJJlC7wEShQNw9UWWg1BWNFjUDsLZ74uxy7k4ung5hA0tRKjDCYoTtKIADP2lZU_gYdSZXkuVBul-SqFl4jZEFRrL6Jm/s320/pro8.gif" alt="" id="BLOGGER_PHOTO_ID_5329446757150735586" border="0" /></a></p><br /><br /><br /><br /></div><p><span style="color: rgb(51, 51, 255); font-weight: bold;">Procedimiento #9:</span><br /></p></div><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjEqhA_Ay0JsknfFJoi7pHQdMn9DCoTzwa-PSRjtZDMmoRh-Ved29QPSrDhzOomY88kM3Qeq62uZjto7SqakiwVl-9LAvAXgZTPmAXWxpAOxavpikBqmA6TIbz1jUU2T956aIvcVjluMP5/s1600-h/pro9.gif"><img id="BLOGGER_PHOTO_ID_5326863417709239954" style="margin: 0px 10px 10px 0px; float: left; width: 120px; height: 45px;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjEqhA_Ay0JsknfFJoi7pHQdMn9DCoTzwa-PSRjtZDMmoRh-Ved29QPSrDhzOomY88kM3Qeq62uZjto7SqakiwVl-9LAvAXgZTPmAXWxpAOxavpikBqmA6TIbz1jUU2T956aIvcVjluMP5/s320/pro9.gif" border="0" /></a><br /><br /></div><br /><br /><br /><span style="color: rgb(51, 51, 255); font-weight: bold;">Procedimiento #10:</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0ftH8wVGEWV8nAzNVoZxXgZkMAFG7fTAcGPMsvwaWUUoWWERisrmPBz2xby1klAZc-YRjwQg8UsFKUAGaVH24wArRJLxrOCvDAaHafKZoo4Q7A1Z6VGpfL4mcVsFX7_SeHnoQCDAf7GWK/s1600-h/pro10.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 124px; height: 49px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0ftH8wVGEWV8nAzNVoZxXgZkMAFG7fTAcGPMsvwaWUUoWWERisrmPBz2xby1klAZc-YRjwQg8UsFKUAGaVH24wArRJLxrOCvDAaHafKZoo4Q7A1Z6VGpfL4mcVsFX7_SeHnoQCDAf7GWK/s320/pro10.gif" alt="" id="BLOGGER_PHOTO_ID_5329447034670209778" border="0" /></a><br /><p>p</p><p><br /></p><p><span style="color: rgb(51, 51, 255); font-weight: bold;">Procedimiento #11: </span><span style="font-weight: bold;"><br /></span></p><p><span style="font-weight: bold;"><br /></span><span style="font-weight: bold;"></span><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguryvIJ9AGr6IgjgSleI6c3XBDAhKe5fjn2Agc7_NdqUwAEMPtxMYR6kcUSNCwTobvRpPUWWwItL_E3R_UWxiOAhPITVUedPGIEo01Y13BabWOkAnkkfADN-IetJM0rMxoBs96B653zYw0/s1600-h/pro11.gif"><img style="margin: 0pt 10px 10px 0pt; float: left; cursor: pointer; width: 241px; height: 47px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguryvIJ9AGr6IgjgSleI6c3XBDAhKe5fjn2Agc7_NdqUwAEMPtxMYR6kcUSNCwTobvRpPUWWwItL_E3R_UWxiOAhPITVUedPGIEo01Y13BabWOkAnkkfADN-IetJM0rMxoBs96B653zYw0/s320/pro11.gif" alt="" id="BLOGGER_PHOTO_ID_5329447264020992578" border="0" /></a></p></div></div></div></div></div></div></div>_Natalia_http://www.blogger.com/profile/01817418121318373428noreply@blogger.com0tag:blogger.com,1999:blog-6250075125315231337.post-44060862737448726192009-04-13T11:50:00.000-07:002009-04-27T12:23:55.254-07:00Sumatoria<div align="center">La <span style="color: rgb(102, 0, 204);"><em>Sumatoria</em> </span>es un tema un tanto difícil de trabajar ya que con todos los elementos de una determinada sucesión.</div><br /><div><br /></div><br /><div>Por lo cúal para lograr facilitar este trabajo se conviene representar la adición de los términos en forma abreviada mediante el signo ∑ , el cuál se acompaña de una fórmula o término general que define a la sucesión y el rango de valores que se tomará en la variable correspondiente.<br /><br /></div><div><br /></div><div align="center">"Por lo que esta se entiende como la suma de un finito de números", denotados de la siguiente manera:</div><div><br /></div><p><span style="color: rgb(0, 102, 0);"><img id="BLOGGER_PHOTO_ID_5326847058232149762" style="margin: 0px auto 10px; display: block; width: 257px; height: 45px; text-align: center;" alt="" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-IUNzABpV4neLUVYU9p9sOJRV8s4qnh0HNdKNlDyI24RSZ84ePy6gK2p0cDh9uP7OILg69vfrRujboUVAZpaWnAMpfdtWjHyGcOT4rJ2SiFbiWfjTybTLLdCBz8qFa7Vj9Yst9wgkPPCA/s320/Image912%5B1%5D.gif" border="0" /></span> </p> <p> </p><p> </p><p><br />Donde:<br /></p><p><span style="color: rgb(255, 0, 0);"> S</span>: Magnitud resultante de la suma.</p><p><span style="color: rgb(255, 0, 0);"> T</span>: Cantidad de valores a sumar.</p><p> <span style="color: rgb(255, 0, 0);">K</span>: Indice de la suma, que va´ría entre h y h+t.</p><div><span style="color: rgb(255, 0, 0);"> H</span>: Punto inicial de la sumatoria.</div><br /><div><span style="color: rgb(255, 0, 0);"> H</span>+<span style="color: rgb(255, 0, 0);">T</span>: punto final de la sumatoria.</div><br /><div><span style="color: rgb(255, 0, 0);"> NK</span>: Valor de la magnitud objeto de suma en el punto k.</div><br /><br /><br /> <span style="font-weight: bold; color: rgb(0, 102, 0);">Donde la notación</span><span style="font-weight: bold; color: rgb(0, 102, 0);">:</span><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0vaVUFh9_RWvHMcOr47r48kf-U7CEZ5dbtHll9z9q6u1XBpHQpiN16flIUE8k5O8xhz7QiDR4RWnB_pWjRPwYDnjxQBjm3IV2qCzX24qr8XM3tIPvqix4JOQTI3XGndhv5oXeJ1Lh3Nac/s1600-h/Image240.gif"><img style="cursor: pointer; width: 42px; height: 52px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0vaVUFh9_RWvHMcOr47r48kf-U7CEZ5dbtHll9z9q6u1XBpHQpiN16flIUE8k5O8xhz7QiDR4RWnB_pWjRPwYDnjxQBjm3IV2qCzX24qr8XM3tIPvqix4JOQTI3XGndhv5oXeJ1Lh3Nac/s320/Image240.gif" alt="" id="BLOGGER_PHOTO_ID_5329450773971175858" border="0" /></a> se lee: <br /><br /> <span style="color: rgb(0, 0, 0);">Suma de X sub-i (ó sigma sub-i) donde i asume todos los valores de 1 hasta n, ó simplemente suma de X sub-i donde i va de 1 a n.</span><br /><br /><span style="color: rgb(0, 0, 128);"><br /> <span style="color: rgb(0, 0, 0);"> Las sumatorias se pueden representar bajo dos tipos de notaciones:</span></span><br /><br /><span style="color: rgb(128, 0, 0);"><span style="color: rgb(153, 51, 153);"> * <span style="font-weight: bold;">Notación suma abierta:</span></span> </span><span style="color: rgb(0, 0, 128);"><span style="color: rgb(0, 0, 0);">Esta notación va de una representación de sumatoria a cada uno de los elementos que la componen, por ejemplo:</span> <img style="color: rgb(0, 0, 102);" src="http://colposfesz.galeon.com/est501/suma/sumahtml/notasuma/Dibujo1.gif" width="164" align="middle" height="53" /></span><span style="color: rgb(128, 0, 0);"> <br /><br /> <span style="font-weight: bold; color: rgb(153, 51, 153);">* Notación suma pertinente:</span><span style="color: rgb(0, 0, 0);"> </span></span><span style="color: rgb(0, 0, 128);"><span style="color: rgb(0, 0, 0);">Esta notación es al contrario de la suma abierta, va de la representación de cada uno de los elementos de una sumatoria a su representación matemática resumida, por ejemplo:</span> <img src="http://colposfesz.galeon.com/est501/suma/sumahtml/notasuma/Dibujo2.gif" width="164" align="middle" height="53" /></span><br /><br /><br /><span style="color: rgb(0, 0, 0);"> </span><br /><span style="color: rgb(51, 0, 51);font-size:130%;" > Ejemplo:</span><span style="font-weight: bold;"> </span><span style="font-weight: bold; color: rgb(0, 0, 128);">Si X<sub>1</sub> = 3 X<sub>2</sub> = 9 X<sub>3</sub> =11</span> <br /><br /><br />encontrar:<br /><br /><img src="http://colposfesz.galeon.com/est501/suma/sumahtml/notasuma/Image242.gif" width="42" align="middle" height="52" /><br /><br /><br /> <span style="font-weight: bold; color: rgb(51, 0, 51);font-size:130%;" > Solución:</span><br /><br /><br /><p align="center"><img src="http://colposfesz.galeon.com/est501/suma/sumahtml/notasuma/Image245.gif" width="182" height="134" /></p><br /><br /><img src="file:///C:/Windows/Temp/moz-screenshot.jpg" alt="" /><br /><div><br /></div><div align="center"><span style="color: rgb(0, 0, 153);font-family:verdana;" >"<span style="font-family:georgia;"><span style="color: rgb(255, 0, 0);">El signo <strong>∑ </strong>corresponde a la letra mayúscula sigma, del alfabeto griego. Es equivalente a la letra<strong> S</strong>, de nuestro alfabeto</span>."</span></span><br /></div><div><br /></div><div><br /></div><br /><div align="left"><br /></div><br /><br /><div align="center"><span style="color: rgb(0, 0, 153);"></span></div><br /><br /><div><br /></div><p align="center"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtXngrs75wfF2cwxgUrGiiEgN8UuIoF4xMn_a0S5Lp0Fa2m90AbjqhUM-htc8KDY-sT9BTa1OGrO6XMlsdbzVL-19lWnw0kJDsKQb-_frFMI42UcgiHbh7sPEe6cc618RLMvxItZ5mrTob/s1600-h/img343%5B1%5D.gif"></a> </p><div><br /></div><br /><br /><div align="left"><br /><br /><br /></div><div align="left"></div><div align="left"></div><div align="left"></div><div align="left"></div>_Natalia_http://www.blogger.com/profile/01817418121318373428noreply@blogger.com3