Math 192—Calculus II Name ___________________________________

Prof. Hess

Summary of Tests of Convergence and Divergence

for Infinite Series

(From Section 12.7)

Name of Test	Convergence or Divergence	Comments
Test for Divergence	If , then San diverges.	Inconclusive if
Integral Test	San and either both converge or both diverge.	Can only be used if f(x) is continuous, positive, and decreasing "x > 1. The antiderivative may be hard to find.
Basic Comparison Test	If Sbn converges and an < bn "n, then San also converges. If Sbn diverges and an > bn "n, then San also diverges.	Can only be used if San and Sbb are both positive terms series. Most useful when San has a form similar to a geometric or p-series.
Limit Comparison Test	If , where c ¹0 and c Î Â, then San and Sbn either both converge or both diverge.	Can only be used if San and Sbb are both positive terms series. Most useful when San has a form similar to a geometric or p-series. Inconclusive if c = 0 or ¥.
Alternating Series Test	Let San = S(-1)^n+1bn. If bn+1 < bn "n, and if , then San converges.	Applies only to alternating series.
Ratio Test	Let . If L < 1, then San converges absolutely. If L > 1, then San diverges.	Inconclusive if L = 1. Most useful when San involves factorials and/or n^th powers.
Root Test	Let . If L < 1, then San converges absolutely. If L > 1, then San diverges.	Inconclusive if L = 1. Most useful when San involves n^th powers.

Common Forms to Memorize

Geometric Series: converges to if |r| < 1, diverges if |r| > 1. To recognize a geometric series, look for a fraction with one or more constants raised to the n^th power.

p-Series: converges if p > 1, diverges if p< 1. To recognize a p-series, look for a fraction with n raised to a constant power. The power may be “hiding” as a radical sign.

Telescoping Series: converges to 1.

Harmonic Series: diverges.

Alternating Harmonic Series: converges conditionally.

Error in Using the Partial Sum S_n to Approximate the Infinite Sum S

For positive term series, (but only if f is continuous, positive, and decreasing).

For alternating series, R_n < b_n+1

Absolute vs Conditional Convergence

Suppose Sa_n is a convergent series that contains both positive and negative terms.

Sa_n converges absolutely if S|a_n| also converges.

Sa_n converges conditionally if S|a_n| diverges.

If you can show that S|a_n| converges, then Sa_n is automatically absolutely convergent.

If S|a_n| diverges, then Sa_n is either divergent or conditionally convergent. Apply the Alternating Series Test to determine which.

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