11.2 - Series

StudyBlue printing of 11.2 - 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; } } Series A series is sum of all the terms in an infinite sequence. We denote this with sigma notation. 00 E an or E an n=1 Just like with sequences, it is very important to talk about convergence and divergence of the infinite series. If the sum of all the terms seems to approach a number as you add an infinite amount of terms, then the series is called convergent. If the sum of the terms seems to get infinitely large as you add an infinite amount of terms, then the series is called divergent. This theory works when you are given the infinite series as terms. If given a sigma notation however, it may be difficult to use this theory. But fear not, because there are different tests that can be used, that will be explained in later parts of chapter 11. Geometric Series One special type of series is the geometric series. This is a series where each term is obtained by multiplying the previous term by a common factor. we express this in sigma notation as 00 E a*r^n-1 = a + a*r + a*r^2 +.... n=1 The reason this is a special series is because we can tell if it converges or diverges just by looking at it. If the absolute value of r is less than 1, than your series is convergent, because this makes the series decrease at a fast enough rate to approach a number. If the absolute value of r is greater than 1, them your series is divergent, because this makes the series get infinitely bigger. Writing it out in math notation, it looks like this if |r|<1, series converges if |r|>1, series diverges this series is even more special, because if it converges we can find what it converges to. The sum of an infinite geometric series is given by 00 E a*r^n-1 = a/(1 - r) as long as |r|<1 n=1 remember, a series is a sum of numbers, so if it is convergent it's sum is a number, while if it is divergent it's sum goes to infinity. Therefore the sum of a geometric series only works when the series converges, which occurs when |r|<1 Limit Test The limit test is a way to test for divergence. It is usually a quick first test that saves a lot of work if it works out. What you do is that when you have the series E an, The you take the limit of an as n approaches 00. If an goes to a number that is not zero, then we know the series diverges, because each of the terms is getting bigger, approaching a non zero constant. IF THE LIMIT GOES TO ZERO, THEN WE KNOW NOTHING! Thats very important, because it tells us that the test failed and we need to move on to another test. in actual math notation it looks like this if lim an (does not equal) 0, then the series diverges. n-->00 if lim an = 0, THEN WE KNOW NOTHING! n-->00 Series manipulations There are several ways to manipulate a series, but they have to be convergent series. If you have the convergent series E an then the series E c*an is also convergent, given c is a constant. You can manipulate series like this by pulling out the constant c, which looks like 00 00 E c*an = c * E an n=1 n=1 Given two convergent series, E an and E bn, it can be said that 00 00 E (an - bn) and E (an+bn) are both convergent series. n=1 n=1 These can be manipulated as well. 00 00 00 E (an - bn) = E an - E bn n=1 n=1 n=1 00 00 00 E (an + bn) = E an + E bn n=1 n=1 n=1 These manipulations can be helpful in determining convergence or divergence, because making the series simpler allows you to perform tests on them or just see something you may of missed when looking at a more complicated expression.