9.2 Tests for Convergence

The Harmonic Series

This series diverges (text does nice proof on pg 721 that continues to increase)

The Divergence Test

If does not exist or if , then diverges.

NOTE: The converse is not necessarily true, as harmonic goes to zero, but sum does not!!

In essence, this says if each term is not close to zero, then the sum “grows”

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The Integral Test Let be a series with for all n. If f(x) is a function such that for all n and if f is continuous, positive, and decreasing for , then and either both converge or both diverge. That is: converges if converges

and diverges if diverges

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The p-Series Test The p-series converges if p > 1 and diverges if p < 1

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(We are omitting the Direct Comparison Test on pg. 727)

The Ratio Test For the series , let . Then:

a) If L < 1, the series converges

b) If L >1, the series diverges

c) If L = 1, the test is inconclusive; it may converges or it may diverge

The ratio test is particularly useful for series that converge rapidly. Series involving factorial or exponential functions are frequently of this type.

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Summary of Tests of Series

Test Series Converges Diverges

Geometric

Divergence

Integral converges diverges

p-Series p > 1 0 < p < 1

Ratio

Challenge – When mixed up – Which Test to use!!

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