# Difference: InfiniteSeriesStrategySheet ( vs. 1)

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# Infinite Series Strategy Sheet

## Know the Famous Series

Did I mention you should know the Famous Series?

## 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 worst, you'll have a great idea how to proceed:

then you don't know what happens. Despair Not! Your work was not in vain.
• Ask yourself: How does go to zero?
• In the limit, does 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 < L < ∞ then the two series behave the same. If L is 0 or ∞ 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

### 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.

-- DickFurnas - 17 Nov 2008

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