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---+ Infinite Series Strategy Sheet
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---++ Know the [[Famous Series]]
Did I mention you should know the [[Famous Series]]?

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---++ Test for Divergence
Unless you immediately recognize a series (_e.g._ it's one of the [[Famous Series]], or an obvious candidate for one of the [[Convergence Tests]]) always start with this one. Why?
---+++ At best, you could be done already!
<latex>\mbox{ \small If $\lim_{k \rightarrow \infty} u_k \ne 0$ then  $\sum u_k$ diverges.}</latex>
---+++ At worst, you'll have a great idea how to proceed:
<latex>\mbox{ \small If $\lim_{k \rightarrow \infty} u_k = 0$}</latex>
    then <latex>\mbox{\small $\sum u_k$}</latex> you don't know what happens. *Despair Not!*  Your work was not in vain.
   * Ask yourself: *How*  does <latex>\mbox{ \small $u_k$}</latex> go to zero?
   * In the limit, does <latex>\mbox{ \small $u_k$}</latex> resemble terms in a famous series?
   * Does the famous series have known convergence properties?
      * If so, the problem series almost certainly has similar convergence properties. Set up a comparison with the famous series. The Limit Comparison Test is usually a good bet here, since you've already been looking at a similar limit.
      * If not, go back and review [[Famous Series]] it's probably there.

Did I mention you should know the [[Famous Series]]?

---++ Limit Comparison Test
If you don't find an easy match to a [[Famous Series]], The _Test for Divergence_ will almost always provide you with a [[Famous Series]] to use with the _Limit Comparison Test._ Set up the ratio between individual terms of the unknown series and the [[Famous Series]] and find the limit, _L_ . If 0 &lt; _L_ &lt; &infin; then the two series behave the same. If _L_ is 0 or &infin; with any luck, your [[Famous Series]] "wins" the limit of the ratio in a useful way:
   * Your unknown series converges if it is clearly smaller than a convergent [[Famous Series]] -- think about it.
   * Your unknown series diverges if it is clearly larger than a divergent [[Famous Series]] -- think about it.

Did I mention you should know the [[Famous Series]]?

---++ Convergence Tests
---+++ What are the various Convergence Tests?
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---+++ Which one should I use?
You have a number of [[Convergence Tests]] available, and most series can be analyzed with more than one of them. The [[Convergence Tests]] page has guidelines for diagnosing when a test is likely to work on a particular series.

-- Main.DickFurnas - 17 Nov 2008