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 Math Help > Sequences and Series > Series > Series Convergence Tests

## Definition of "Converge"

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

limn—> Sn = S,

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

A series ak is said to converge if the sequence Sn of partial sums converges.

Note:
 For j ≥ 0, ∞ ∑ k=0 ak converges iff ∞ ∑ k=j ak converges,

## 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 kp for various values of p.  If p < -1 then 1xp = -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 1x-1 doesn't converge, so k-1 doesn't converge.

### Comparison test

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

1. If bk converges, then ak also converges, and
2. If ak diverges, then bk also diverges.

### Limit comparison test

Let ak and bk be series with positive terms.
 If lim k—>∞ ak — bk = L
where 0 < L < ∞, then ak and bk either both converge or both diverge.

### Ratio test

Let ak be a series with positive terms, and suppose that
 lim k—>∞ ak+1 —— ak = L
1. If L < 1, then ak converges.
2. If L > 1, then ak 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 ak be a series with non-negative terms, and suppose that
 lim k—>∞ (ak)1/k = L
1. If L < 1, then ak converges.
2. If L > 1, then ak 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.