Testing Convergence at Endpoints

Integral Test:

If f is positive, continuous, and decreasing for x > N (a positive integer) and a_n=f(n),

then
¥
å     a_n
n=N                 and                  ¥
                                               ò     f(x) dx
                                               N

either both converge or both diverge.

p-series test:

A series of the form

¥
å 1 / n ^p = 1 + 1/2 ^p + 1/3 ^p + 1/4 ^p + ....
n=1

is called a p-series, where p is a positive constant. For p = 1, the series is the harmonic series.

Limit Comparison Test:

Suppose a_n > 0 and b_n > 0 for all n > N, and c is finite and positive. Then if

1. lim a_n = c both series converge
n®¥ b_n or both diverge

2. lim a_n = 0 If the b series converges,
n®¥ b_n then the a series converges

3. lim a_n = ¥ If the b series diverges,
n®¥ b_n then the a series diverges

Alternating Series Test (Liebniz’s Theorem):

The series

¥
å (-1)ⁿ⁺¹a_n = a₁ - a₂ + a₃ - a₄ + ...
n=1

converges if all below are true
1. Each u_n is positive
2. u_n > u_n+1 for all n > N
3. lim a_n = 0
n®¥

Alternating Series Estimation Theorem:

                                     ¥
If the alternating series   å     (-1)ⁿ⁺¹a_n    =    a₁ - a₂ + a₃ - a₄ + ...
                                   n=1

satisfies the conditions of Liebniz’s Theorem, then the truncation error for the nth partial sum is less than u_n+1 and has the same sign as the unused term.

Rearrangements of Series:

If the series a_n converges absolutely, and if b₁, b₂, …, is any rearrangement of the sequence a_n, then the series b_n converges absolutely and the two sums are equal.

If the series a_n converges conditionally, then the terms can be rearranged to form a divergent series.
The terms can also be rearranged to form a series that converges to any preassigned sum.