Definition of "Converge"

A sequence S_n converges to the limit S -- that is,

lim_n—> S_n = S,

-- if for any positive real number, ε, there exists a positive integer, N, such that
n > N  ==>  |S_n - S| < ε   

A series ∑a_k is said to converge if the sequence S_n of partial sums converges.

Note:

For j ≥ 0,

∞ ∑ k=0

ak converges iff

∞ ∑ k=j

ak converges,

So for the discussion on this page, I'll write ∑a_k.

Convergence tests

Divergence test

If

lim k—>∞

ak ≠ 0, then ∑ak. diverges.

Integral test

Let f(x) be continuous, decreasing, and positive for x ≥ 1.

Then

∞ ∑ k=1

f(k) converges if and only if ∫

∞ 1

f(x)dx converges.

Example:

Consider ∑k^p for various values of p.  If p < -1 then ∫₁^∞x^p = -1/(p+1), but otherwise, the integral doesn't converge, so neither does the series.  In particular, if p=1, then ∫x^-1 = ln|x|, so ∫₁ ^∞x^-1 doesn't converge, so ∑k^-1 doesn't converge.

Comparison test

Let ∑a_k and ∑b_k be series with non-negative terms.  If a_k ≤ b_k for all k sufficiently large, then:

1. If ∑b_k converges, then ∑a_k also converges, and
2. If ∑a_k diverges, then ∑b_k also diverges.

Limit comparison test

Let ∑a_k and ∑b_k be series with positive terms.

If

lim k—>∞

ak — bk

= L

where 0 < L < ∞, then ∑a_k and ∑b_k either both converge or both diverge.

Ratio test

Let ∑a_k be a series with positive terms, and suppose that

lim k—>∞

ak+1 —— ak

= L

1. If L < 1, then ∑a_k converges.
2. If L > 1, then ∑a_k diverges.
3. If L = 1, then the test is inconclusive.

A geometric series converges if its common ratio is strictly between -1 and 1.  For 0 < r < 1, the ratio test shows convergence.  For -1 < r < 0, the alternating series test (below) shows convergence.  For r=0 the comparison test with ∑2^-k shows convergence.

For -1 < r < 1,

∞ ∑ k=0

ark  =

a ——— (1-r)

Raabe's test

If the ratio test is inconclusive (L=1) then if

lim k—>∞

n(

ak+1 —— ak

-1) < -1

then the series converges by Raabe's test.

 . . . . . . an example should be added here to show how Raabe's test can be used.  Some resources say the hypergeometric series is an example that can be shown to converge using this test.

Root test

Let ∑a_k be a series with non-negative terms, and suppose that

lim k—>∞

(ak)1/k

= L

1. If L < 1, then ∑a_k converges.
2. If L > 1, then ∑a_k diverges.
3. If L = 1, then the test is inconclusive.

Alternating series test

The alternating series

∞ ∑ k=0

(-1)kak

converges if ak+1 < ak for all k and

lim k—>∞

ak

= 0.

Internet references

Convergent Sequence, in Mathworld

Harvey Mudd College Calculus Tutorial: Convergence tests for infinite series 

Related pages in this website

Calculus Theorems -- important facts about continuous functions

The webmaster and author of this Math Help site is Graeme McRae.