https://wiki.math.ucr.edu/index.php?title=Strategies_for_Testing_Series&feed=atom&action=history Strategies for Testing Series - Revision history 2022-01-21T09:14:03Z Revision history for this page on the wiki MediaWiki 1.35.0 https://wiki.math.ucr.edu/index.php?title=Strategies_for_Testing_Series&diff=1684&oldid=prev MathAdmin: Created page with "In general, there are no specific rules as to which test to apply to a given series. Instead, we classify series by their form and give tips as to which tests should be cons..." 2017-10-30T18:48:00Z <p>Created page with &quot;In general, there are no specific rules as to which test to apply to a given series. Instead, we classify series by their form and give tips as to which tests should be cons...&quot;</p> <p><b>New page</b></p><div>In general, there are no specific rules as to which test to apply to a given series. <br /> <br /> Instead, we classify series by their form and give tips as to which tests should be considered. <br /> <br /> This list is meant to serve as a guideline for which tests you should consider applying to a given series.<br /> <br /> '''1.''' If the series is of the form <br /> <br /> :::&lt;math style=&quot;vertical-align: -10px&quot;&gt;\sum \frac{1}{n^p} &lt;/math&gt;&amp;nbsp; or &amp;nbsp;&lt;math style=&quot;vertical-align: -5px&quot;&gt;\sum ar^n,&lt;/math&gt;&amp;nbsp; <br /> <br /> :then the series is a &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;p-&lt;/math&gt;series or a geometric series<br /> <br /> :For the &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;p-&lt;/math&gt;series <br /> <br /> :::&lt;math&gt;\sum \frac{1}{n^p},&lt;/math&gt;&amp;nbsp; <br /> <br /> :it is convergent if &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;p&gt;1&lt;/math&gt;&amp;nbsp; and divergent if &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;p\le 1.&lt;/math&gt;<br /> <br /> :For the geometric series <br /> <br /> :::&lt;math&gt;\sum ar^n,&lt;/math&gt;&amp;nbsp; <br /> <br /> :it is convergent if &amp;nbsp;&lt;math style=&quot;vertical-align: -5px&quot;&gt;|r|&lt;1&lt;/math&gt;&amp;nbsp; and divergent if &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;|r|\ge 1.&lt;/math&gt;<br /> <br /> '''2.''' If the series has a form similar to a &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;p-&lt;/math&gt;series or a geometric series, <br /> <br /> :then one of the comparison tests should be considered.<br /> <br /> '''3.''' If you can see that <br /> <br /> :::&lt;math&gt;\lim_{n\rightarrow \infty} a_n \neq 0,&lt;/math&gt;&amp;nbsp; <br /> <br /> :then you should use the Divergence Test or &amp;nbsp;&lt;math style=&quot;vertical-align: 0px&quot;&gt;n&lt;/math&gt;th term test.<br /> <br /> '''4.''' If the series has the form <br /> <br /> :::&lt;math style=&quot;vertical-align: -6px&quot;&gt;\sum (-1)^n b_n&lt;/math&gt;&amp;nbsp; or &amp;nbsp;&lt;math style=&quot;vertical-align: -6px&quot;&gt;\sum (-1)^{n-1} b_n&lt;/math&gt;&amp;nbsp; <br /> <br /> :with &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;b_n&gt;0&lt;/math&gt;&amp;nbsp; for all &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;n,&lt;/math&gt;&amp;nbsp; then the Alternating Series Test should be considered.<br /> <br /> '''5.''' If the series involves factorials or other products, the Ratio Test should be considered. <br /> <br /> :&lt;u&gt;NOTE:&lt;/u&gt; The Ratio Test should not be used for rational functions of &amp;nbsp;&lt;math style=&quot;vertical-align: 0px&quot;&gt;n.&lt;/math&gt;&amp;nbsp; <br /> <br /> :For rational functions, you should use the Limit Comparison Test.<br /> <br /> '''6.''' If &amp;nbsp;&lt;math style=&quot;vertical-align: -5px&quot;&gt;a_n=f(n)&lt;/math&gt;&amp;nbsp; for some function &amp;nbsp;&lt;math style=&quot;vertical-align: -5px&quot;&gt;f(x)&lt;/math&gt;&amp;nbsp; where <br /> <br /> :::&lt;math&gt;\int_a^\infty f(x)~dx&lt;/math&gt;&amp;nbsp; <br /> <br /> :is easily evaluated, the Integral Test should be considered. <br /> <br /> '''7.''' If the terms of the series are products involving powers of &amp;nbsp;&lt;math style=&quot;vertical-align: -4px&quot;&gt;n,&lt;/math&gt;&amp;nbsp; <br /> <br /> :then the Root Test should be considered.<br /> <br /> &lt;u&gt;NOTE:&lt;/u&gt; These strategies are used for determining whether a series converges or diverges. <br /> <br /> However, these are not the strategies one should use if we are determining whether or not a <br /> <br /> series is absolutely convergent.</div> MathAdmin