{"id":10713,"date":"2017-09-21t11:24:23","date_gmt":"2017-09-21t18:24:23","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=10713"},"modified":"2017-09-21t11:24:45","modified_gmt":"2017-09-21t18:24:45","slug":"ap-calculus-bc-review-absolute-conditional-convergence","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-review-absolute-conditional-convergence\/","title":{"rendered":"ap calculus bc review: absolute and conditional convergence"},"content":{"rendered":"<p>any given series either <em>converges<\/em> or <em>diverges<\/em>.  but did you realize that there were different kinds of convergence?  in this review article, we&#8217;ll take a look at the difference between absolute and conditional convergence.  along the way, we&#8217;ll see a few examples and discuss important special cases.<\/p>\n<h2>absolute and conditional convergence<\/h2>\n<p>if a series has a finite sum, then the series <strong>converges<\/strong>.  otherwise, the series <strong>diverges<\/strong>.<\/p>\n<p>however, we do make a distinction between series that converge very strongly, and those that only <em>just barely<\/em> converge.  it all boils down to the signs of the terms.<\/p>\n<ul>\n<li>\na series &sigma;<em>a<sub>n<\/sub><\/em> converges <strong>absolutely<\/strong> if the series of the absolute values of its terms converges.  that is, if &sigma;|<em>a<sub>n<\/sub><\/em>| also converges.<\/li>\n<li>a series &sigma;<em>a<sub>n<\/sub><\/em> converges <strong>conditionally<\/strong> if the series converges but the associated series of absolute values &sigma;|<em>a<sub>n<\/sub><\/em>| diverges.<\/li>\n<\/ul>\n<h3>classifying series<\/h3>\n<p>it&#8217;s important to realize that both absolute and conditional convergence are still types of convergence.  it&#8217;s not that a conditionally convergent series <em>sometimes<\/em> converges and sometimes not.  instead, we&#8217;re talking about the behavior of a related series: <em>what happens when we get rid of all the negative signs in the series?<\/em>  if the new series of all positive terms <em>also<\/em> converges, then the original series converges in the strongest possible sense (absolute convergence).<\/p>\n<p>so, there are only three possibilities:<\/p>\n<ol>\n<li>a series could converge absolutely.<\/li>\n<li>it could converge conditionally.<\/li>\n<li>or, the series could diverge.<\/li>\n<\/ol>\n<figure id=\"attachment_10053\" aria-describedby=\"caption-attachment-10053\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/05\/yes-945356_960_720-e1496247159482.jpg\" alt=\"conditional convergence is not quite yes, no, or maybe\" width=\"500\" height=\"407\" class=\"size-full wp-image-10053\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/05\/yes-945356_960_720-e1496247159482.jpg 500w, \/\/www.catharsisit.com\/hs\/files\/2017\/05\/yes-945356_960_720-e1496247159482-300x244.jpg 300w, \/\/www.catharsisit.com\/hs\/files\/2017\/05\/yes-945356_960_720-e1496247159482-30x24.jpg 30w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-10053\" class=\"wp-caption-text\">conditional convergence is not quite <em>yes<\/em>, <em>no<\/em>, or <em>maybe<\/em>.  instead, it&#8217;s more like <em>yes definitely<\/em> (absolute convergence), <em>just barely yes<\/em> (conditional convergence), or <em>no<\/em> (divergence).<\/figcaption><\/figure>\n<h2>prototypical examples<\/h2>\n<p>let&#8217;s talk about three basic series to help illustrate the point.<\/p>\n<p>usually the question of absolute versus conditional convergence pertains to series that have a mixture of positive and negative terms &#8212; as in an alternating series.  (you can check out this review article for more details about alternating series: <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-review-alternating-series\/\">ap calculus bc review: alternating series<\/a>.)<\/p>\n<h3>example 1 &#8212; absolutely convergent<\/h3>\n<p>first, let&#8217;s consider the following series.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/alternating_geometric.gif\" alt=\"alternating geometric series\" width=\"220\" height=\"95\" class=\"aligncenter size-full wp-image-10946\" \/><\/p>\n<p>this series is both <em>alternating<\/em> (the signs switch back and forth) and <em>geometric<\/em> (there is a common ratio).  in fact, because the common ratio, <em>r<\/em> = -1\/2, has absolute value less than 1, we know that this series converges.<\/p>\n<p>but is the convergence absolute or conditional?<\/p>\n<p>consider the related series of absolute values of each term:<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/absolute_series.gif\" alt=\"series of absolute values\" width=\"219\" height=\"154\" class=\"aligncenter size-full wp-image-10947\" \/><\/p>\n<p>again, we have a <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-review-geometric-series\/\">geometric series<\/a>.  but this time, the ratio is positive: <em>r<\/em> = 1\/2.  still, |<em>r<\/em>| &lt; 1, so the series of absolute values of terms converges as well.<\/p>\n<p>therefore, we have shown that the original series is <strong>absolutely convergent<\/strong>.<\/p>\n<p><strong>caution:<\/strong> the two series (original vs. absolute values) are not usually going to converge to the same value.  indeed, using the formula for the sum of a geometric series, we can find out the sums explicitly:<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/comparing_sums.gif\" alt=\"sum of the original series is 2\/3.  sum of the absolute values is 2.\" width=\"340\" height=\"94\" class=\"aligncenter size-full wp-image-10948\" \/><\/p>\n<h3>example 2 &#8212; conditionally convergent<\/h3>\n<p>next, consider the <strong>alternating harmonic series<\/strong>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/alternating_harmonic.gif\" alt=\"alternating harmonic series\" width=\"245\" height=\"95\" class=\"aligncenter size-full wp-image-10938\" \/><\/p>\n<p>this series does converge.  how can we tell?  use the alternating series test.<\/p>\n<p>the absolute value of the general term is equal to 1\/<em>n<\/em>, which decreases and limits on the value 0 (as <em>n<\/em> &rarr; &infin;).  this implies that the original alternating series is convergent.<\/p>\n<p>but how convergent is it really?<\/p>\n<p>the series of absolute values is famous.  it&#8217;s called the <strong>harmonic series<\/strong>, and we know that it diverges!<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/harmonic_series.gif\" alt=\"harmonic series\" width=\"300\" height=\"51\" class=\"aligncenter size-full wp-image-10369\" \/><\/p>\n<p>therefore, by definition, the alternating harmonic series is <strong>conditionally convergent<\/strong>.<\/p>\n<h3>example 3 &#8212; divergent<\/h3>\n<p>finally, let&#8217;s take a look at the series of powers of -1.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/sum_alternating.gif\" alt=\"series of powers of -1\" width=\"311\" height=\"77\" class=\"aligncenter size-full wp-image-10880\" \/><\/p>\n<p>this series has no chance of converging at all because the limit of the general term is not zero.  in fact, the limit of (-1)<em><sup>n<\/sup><\/em> as <em>n<\/em> &rarr; &infin; doesn&#8217;t exist at all!<\/p>\n<p>therefore, this series is <strong>divergent<\/strong>.  there is no need to check the sum of absolute values at all.<\/p>\n<h2>the <em>p<\/em>-series test and conditional convergence<\/h2>\n<p>there is a very important class of series called the <em>p<\/em>-series.<\/p>\n<p>by definition, a <strong><em>p<\/em>-series<\/strong> is a series of the form:<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/p-series.gif\" alt=\"p-series\" width=\"48\" height=\"51\" class=\"aligncenter size-full wp-image-10950\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/p-series.gif 48w, \/\/www.catharsisit.com\/hs\/files\/2017\/07\/p-series-28x30.gif 28w\" sizes=\"(max-width: 48px) 100vw, 48px\" \/><\/p>\n<p>there is an extremely simple test for convergence.<\/p>\n<ul>\n<li>if <em>p<\/em> &gt; 1, then the series converges.<\/li>\n<li>if <em>p<\/em> &le; 1, then the series diverges.<\/li>\n<\/ul>\n<p>for example, the harmonic series, &sigma;1\/<em>n<\/em> is a <em>p<\/em>-series with <em>p<\/em> = 1.  according to the chart above, this series must diverge.  <\/p>\n<p>on the other hand, the series of reciprocals of squares, &sigma;1\/<em>n<\/em><sup>2<\/sup> converges because it&#8217;s a <em>p<\/em>-series with <em>p<\/em> = 2, and 2 is greater than 1.<\/p>\n<h3>alternating <em>p<\/em>-series<\/h3>\n<p>but what about the alternating versions of the <em>p<\/em>-series?<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/alternating_p-series.gif\" alt=\"alternating p-series\" width=\"198\" height=\"96\" class=\"aligncenter size-full wp-image-10951\" \/><\/p>\n<p>this time, the answer depends on the alternating series test.  as long as <em>p<\/em> &gt; 0, then there will be a positive power of <em>n<\/em> in the denominator.  so we would have 1\/<em>n<sup>p<\/sup><\/em> &rarr; 0 as <em>n<\/em> &rarr; &infin;, which in turn proves that the alternating series converges.<\/p>\n<p>however, the convergence is only conditional if <em>p<\/em> &le; 1 because of the <em>p<\/em>-series test.  absolute convergence is guaranteed when <em>p<\/em> &gt; 1, because then the series of absolute values of terms would converge by the <em>p<\/em>-series test.<\/p>\n<p>to summarize, the convergence properties  of the alternating <em>p<\/em>-series are as follows.<\/p>\n<ul>\n<li>if <em>p<\/em> &gt; 1, then the series converges absolutely.<\/li>\n<li>if 0 &lt; <em>p<\/em> &le; 1, then the series converges conditionally.<\/li>\n<li>if <em>p<\/em> &le; 0, then the series diverges.<\/li>\n<\/ul>\n<h2>more about absolute convergence<\/h2>\n<p>there are two powerful convergence tests that can determine whether a series is absolutely convergent: the <strong>ratio test<\/strong> and <strong>root test<\/strong>.  however, neither one can tell you about conditional convergence.<\/p>\n<p>in fact, if the series is only conditionally convergent, then both the ratio and root test will turn out to be <em>inconclusive<\/em>. <\/p>\n<p>for example, let&#8217;s try the ratio test on the alternating harmonic series.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/ratio_test_alternating_harmonic.gif\" alt=\"the ratio test for the alternating harmonic series turns out to be inconclusive\" width=\"195\" height=\"169\" class=\"aligncenter size-full wp-image-10949\" \/><\/p>\n<p>because the limit is 1, the ratio test cannot determine whether the series converges or diverges.<\/p>\n<p>you can find out more about these two tests here: <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-review-ratio-root-tests\/\">ap calculus bc review: ratio and root tests<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>did you realize that there were different kinds of series convergence? in this review article, find out about absolute and conditional convergence.<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"class_list":["post-10713","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>ap calculus bc review: absolute and conditional convergence - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"did you realize that there were different kinds of series convergence? in this review article, find out about absolute and conditional convergence.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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