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Summary of Series Convergence Tests


A. Particular Series

(1) Geometric Series


converges to


if |c| < 1,

diverges if |c| ≥ 1.

(2)    Harmonic Series

09_infinite_series-367.gif diverges.

(3)    p Series

09_infinite_series-368.gif converges if p > 1, diverges if p ≤ 1.

B.     Tests for Positive and Alternating Series In the tests below, assume an ≥ 0 for all n.

(1)     Convergence versus Divergence to ∞ Let H be infinite.

09_infinite_series-369.gif converges if 09_infinite_series-370.gif is finite,

diverges to ∞ if 09_infinite_series-371.gif is infinite.

(2)    Comparison Test Suppose an ≤ cbn for all n.

If 09_infinite_series-372.gif converges then 09_infinite_series-373.gifconverges.

If 09_infinite_series-374.gif diverges then 09_infinite_series-375.gifdiverges.

Hint: Often a series can be compared with one of the particular series above: a geometric, harmonic, or p series.

(3)    Limit Comparison Test

Suppose aK ≤ cbK for all infinite K.

If 09_infinite_series-376.gif converges then 09_infinite_series-377.gif converges.

If 09_infinite_series-378.gif diverges then 09_infinite_series-379.gif diverges.

Hint: Try this test if the Comparison Test almost works.

(4)    Integral Test

Suppose f is continuous, decreasing, and positive for x ≥ 1.

If 09_infinite_series-380.gif converges, then 09_infinite_series-381.gif converges.

If 09_infinite_series-382.gif diverges, then 09_infinite_series-383.gif diverges.

Hint: This test may be useful if an comes from a continuous function f(x).

(5)    Alternating Series Test

09_infinite_series-384.gif converges if the an are decreasing and approach 0. Hint: This is usually the simplest test if you see a ( - 1)n in the expression.

C.     Tests for General Series

(1)    Definition of Convergence

09_infinite_series-385.gif converges if and only if the partial sum series09_infinite_series-386.gif converges.

(2)    Cauchy Convergence Test

09_infinite_series-387.gif converges if for all infinite H and K > H, aH+1 + ... + aK ≈ 0,

diverges if for some infinite H and K > H,

aH+1 + ... + aK not amost equal to 0, diverges if limn→∞ an ≠ 0.

Hint: This test is useful for showing a series diverges.

(3)    Constant and Sum Rules

Sums and constant multiples of convergent series converge.

(4)    Tail Rule

09_infinite_series-388.gif converges if and only if 09_infinite_series-389.gif converges.

(5)    Absolute Convergence

If 09_infinite_series-390.gif converges then 09_infinite_series-391.gif converges.

Hint: Remember that 09_infinite_series-392.gif is a positive term series.

Thus tests in group B may be applied to 09_infinite_series-393.gif

(6)    Ratio Test

Suppose 09_infinite_series-394.gif


converges absolutely if L < 1, diverges if L > 1.

Hint: This is useful if an involves a factorial. Watch for 09_infinite_series-396.gif in the ratio because09_infinite_series-397.gif

If the limit L is one, try another test because the Ratio Test gives no information.

Last Update: 2006-11-07