9.5 | Alternating Series


Alternating Series Test
Let \(a_n > 0\). The alternating series
$$
\sum_{n=1}^\infty (-1)^n a_n
$$
and
$$
\sum_{n = 1}^\infty (-1)^{n+1} a_n
$$
converge when the two conditions listed below are met.

  1. \(\displaystyle \lim_{n \rightarrow \infty} a_n = 0 \)
  2. \(a_{n+1} \leq a_n\), for all \(n\)
Absolute Convergence
If the series \(\sum |a_n|\) converges, then the series \(\sum a_n\) also converges.
Definitions of Absolute and Conditional Convergence

  1. The series \(\sum a_n\) is absolutely convergent when \(\sum|a_n|\) converges.
  2. The series \(\sum a_n\) is conditionally convergent when \(\sum a_n\) converges but \(\sum |a_n|\) diverges.

E 9.5 Exercises