Question

How do you use the Alternating Series Test?

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 apply the test to the alternating series below.

`sum_{n=1}^infty(-1)^{n-1}1/sqrt{n}`

In this series, `b_n=1/sqrt{n}`.

Let us check the two conditions.

1. `1/sqrt{n} ge 1/sqrt{n+1}` for all `n ge 1`
2. `lim_{n to infty}1/n=1/infty=0`

Hence, we conclude that the alternating series converges.

I hope that this was helpful.