The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.  ## 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 ≤ cbn 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 aK ≤ cbK 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 an comes from a continuous function f(x).

(5)    Alternating Series Test 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 converges if and only if the partial sum series converges.

(2)    Cauchy Convergence Test 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 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 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 an 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.

Last Update: 2006-11-07