Question

What is the Alternating Series Test of convergence?

Answer 1

Alternating Series Test states that an alternating series of the form
`sum_{n=1}^infty (-1)^nb_n`, where `b_n ge0`,
converges if the following two conditions are satisfied:
1. `b_n ge b_{n+1}` for all `n ge N`, where `N` is some natural number.
2. `lim_{n to infty}b_n=0`

Let us look at the alternating harmonic series `sum_{n=1}^infty (-1)^{n-1}1/n`.
In this series, `b_n=1/n`. Let us check the two conditions.
1. `1/n ge 1/{n+1}` for all `n ge 1`
2. `lim_{n to infty}1/n=0`

Hence, we conclude that the alternating harmonic series converges.