Summary of Series Convergence Tests

SUMMARY OF SERIES CONVERGENCE TESTS

A. Particular Series

(1) Geometric Series

converges to

if |c| < 1,

diverges if |c| ≥ 1.

(2)    Harmonic Series

diverges.

(3)    p Series

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.

converges if is finite,

diverges to ∞ if is infinite.

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

If converges then converges.

If diverges then diverges.

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 a_K ≤ cb_K for all infinite K.

If converges then converges.

If diverges then 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 converges, then converges.

If diverges, then diverges.

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

(5)    Alternating Series Test

converges if the a_n 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

converges if and only if the partial sum series converges.

(2)    Cauchy Convergence Test

converges if for all infinite H and K > H, a_H+1 + ... + a_K ≈ 0,

diverges if for some infinite H and K > H,

a_H+1 + ... + a_K 0, diverges if lim_n→∞ a_n ≠ 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

converges if and only if converges.

(5)    Absolute Convergence

If converges then converges.

Hint: Remember that is a positive term series.

Thus tests in group B may be applied to

(6)    Ratio Test

Suppose

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

Hint: This is useful if a_n involves a factorial. Watch for in the ratio because

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