11.8 - Power Series.

StudyBlue printing of 11.8 - Power Series. html, body, div, span, applet, object, iframe, h1, h2, h3, h4, h5, h6, p, blockquote, pre, a, abbr, acronym, address, big, cite, code, del, dfn, em, font, img, ins, kbd, q, s, samp, small, strike, strong, sub, sup, tt, var, b, u, i, center, fieldset, form, label, legend, table, caption, tbody, tfoot, thead, tr, th, td { margin: 0; padding: 0; border: 0; outline: 0; font-size: 100%; background: transparent; } body { line-height: 1; } blockquote, q { quotes: none; } blockquote:before, blockquote:after, q:before, q:after { content: ''; content: none; } /* remember to define focus styles! */ :focus { outline: 0; } /* remember to highlight inserts somehow! */ ins { text-decoration: none; } del { text-decoration: line-through; } /* tables still need 'cellspacing="0"' in the markup */ table { border-collapse: collapse; border-spacing: 0; } /* end RESET */ .header { min-width:800px; } .logo { padding:6px 20px 2px 20px; margin:0; font-size:25px; font-weight:bold; color:#808285; position:relative; border-bottom: 1px solid #c5c5c5; } .logo-blue { color:#70adc4; } .logo-desc { font-weight:normal; font-size:19px; color:#cccccc; margin-top:50px; position:absolute; display: none; } .back-button { position:absolute; top:20px; right:20px; font-size:13px; line-height:25px; color:rgb(0,175,225); font-weight:normal; } .back-button a { color:rgb(0,175,225); } .instructions { padding:0; margin:0; width:100%; position:relative; color:rgb(100,100,100); } .step-holder { border-left:1px solid #ededed; margin-left:20px; } .steps { padding:15px 0; float:left; width:24%; border-right:1px solid #ededed; text-align:center; } .steps-01 { } .steps-02 { } .steps-03 { } .steps-04 { } .label { padding:5px 10px; } .print-button { } .print-button a { background-color:rgb(0,175,225); color:white; line-height: 19px; padding:9px 8px 5px 30px; font-size:14px; text-decoration:none; background-image: url(images/printer.png); background-repeat: no-repeat; background-position: 7px 50%; -moz-border-radius: 5px; -webkit-border-radius: 5px; } .print-button a:hover { background-color:black; } .theNote .content { width: 8.0in !important; margin: 5px auto; padding:20px; background-color:white; } .theNote .header { border-bottom: 1px dashed #C8C8C8; font-size: 17px; padding: 0 0 10px; line-height: 19px; color: #00ADE1; min-width:500px; } .theNote .body { font-size: 14px; line-height: 19px; padding: 10px 0; } .theNote{ padding:6px 0; clear:both; background-color: rgb(200,200,200); } .theNote h3{ color: rgb(100,100,100); } .theNote h1, .theNote h3{ background-color:white; padding:2px 20px; width:8.0in !important; margin: 0 auto; font-size: 15px; } .theNote h1{ padding-top: 10px; font-size: 15px; } .theNote h1:first-child{ font-size: 20px; } .theNote h3 { font-size: 14px; font-weight: normal; } #options { border: 3px double #ccc; padding: 5px 12px; margin: 10px 50px 10px 20px; float: left; } #info { border-top: 1px solid #ccc; padding-top: 5px; font-style: italic; } li { margin: 5px 10px 5px 25px; } ul li { list-style: disc; } ol li { list-style: decimal; } img { border: 0; } table { clear: both; width: 100%; border: 1px solid #c5c5c5; border-width: 1px 0; margin: 0; page-break-after: always; } table#page { page-break-after: auto; } td { text-align: center; font-size: 12px; border-bottom: 1px dashed #c5c5c5; height: 1.75in; width: 50%; padding-left: 15px; } .leftside { border-right: 1px solid #cccccc; padding: 0 15px 0 0; } .bottom td { border-bottom: none; } .clearfix { clear:both; line-height:1px; height:1px; } img { max-width:80%; max-height:150px; margin:20px; } @media print {.header { display: none; } .content .header{ display:inherit; } table { border: 1px dashed #bbb; border-width: 1px 0; } .theNote{ background-color:white; } } Power Series A power series is any series that follows the form OO E cn*x^n = c0 + c1x + c2 x^2 + c3 x^3..... n=0 Where x is the variable and the c terms are the coefficients. Now the above formula desribes a power series centered around zero. if we wanted to center the power sereis around some other number, a, we would use the following OO E cn*(x-a)^n = c0 + c1(x-a) + c2 (x-a)^2 + c3 (x-a)^3..... n=0 Now what happens if we want to know when a power series converges? Well luckily there are only three possibilities for a power series, and they are 1. The power series only converges when x=a 2 The power series converges for all values of x 3. There is a positive number R such that the series converges if |x-a|<R and diverges for |x-a|>R Possibilities 1 and 2 are self explanatory, but 3 is a little tricker. R is called the radius of convergence. I is then the interval of convergence, which is the interval of x over which the power series converges.