Definition of Convergence and Divergence in Series

The n^th partial sum of the series  a_n is given by S_n = a₁ + a₂ + a₃ + … + a_n. If the sequence of these partial sums {S_n} converges to L, then the sum of the series converges to L. If {S_n} diverges, then the sum of the series diverges.

Operations on Convergent Series

If  a_n = A, and  b_n = B, then the following also converge as indicated:

 ca_n = cA
 (a_n + b_n) = A + B
 (a_n – b_n) = A – B

Alphabetical Listing of Convergence Tests

Absolute Convergence

If the series  |a_n| converges, then the series  a_n also converges.

Alternating Series Test

If for all n, a_n is positive, non-increasing (i.e. 0 < a_n+1 <= a_n), and approaching zero, then the alternating series
 (-1)ⁿ a_n   and    (-1)^n-1 a_n
both converge.
If the alternating series converges, then the remainder R_N = S – S_N (where S is the exact sum of the infinite series and S_N is the sum of the first N terms of the series) is bounded by |R_N| <= a_N+1

Deleting the first N Terms

If N is a positive integer, then the series

an and

an   n=N+1

both converge or both diverge.

Direct Comparison Test

If 0 <= a_n <= b_n for all n greater than some positive integer N, then the following rules apply:
If  b_n converges, then  a_n converges.
If  a_n diverges, then  b_n diverges.

Geometric Series Convergence

The geometric series is given by
 a rⁿ = a + a r + a r² + a r³ + …
If |r| < 1 then the following geometric series converges to a / (1 – r).

If |r| >= 1 then the above geometric series diverges.

Integral Test

If for all n >= 1, f(n) = a_n, and f is positive, continuous, and decreasing then  

an and

an

either both converge or both diverge.
If the above series converges, then the remainder R_N = S – S_N (where S is the exact sum of the infinite series and S_N is the sum of the first N terms of the series) is bounded by 0< = R_N <= (N..) f(x) dx.

Limit Comparison Test

If lim (n–>) (a_n / b_n) = L,
where a_n, b_n > 0 and L is finite and positive,
then the series  a_n and  b_n either both converge or both diverge.

n^th-Term Test for Divergence

If the sequence {a_n} does not converge to zero, then the series  a_n diverges.

p-Series Convergence

The p-series is given by
 1/n^p = 1/1^p + 1/2^p + 1/3^p + …
where p > 0 by definition.
If p > 1, then the series converges.
If 0 < p <= 1 then the series diverges.

Ratio Test

If for all n, n  0, then the following rules apply:
Let L = lim (n — > ) | a_n+1 / a_n |.
If L < 1, then the series  a_n converges.
If L > 1, then the series  a_n diverges.
If L = 1, then the test in inconclusive.

Root Test

Let L = lim (n — > ) | a_n |^1/n.
If L < 1, then the series  a_n converges.
If L > 1, then the series  a_n diverges.
If L = 1, then the test in inconclusive.

Taylor Series Convergence

If f has derivatives of all orders in an interval I centered at c, then the Taylor series converges as indicated:
 (1/n!) f⁽ⁿ⁾(c) (x – c)ⁿ = f(x)
if and only if lim (n–>) RN = 0 for all x in I.
The remainder R_N = S – S_N of the Taylor series (where S is the exact sum of the infinite series and S_N is the sum of the first N terms of the series) is equal to (1/(n+1)!) f⁽ⁿ⁺¹⁾(z) (x – c)ⁿ⁺¹, where z is some constant between x and c.