Question

How to determine is `sum(-1)^n n/(n+1)` absolutely convergent, conditionally convergent or neither?

Support your answer.

Answer 1

neither

Explanation:

`lim_(nrarroo)(-1)^n*n/(n+1) != 0`

diverges by nth term test

Answer 2

Diverges

Explanation:

This is an alternating series. Let us first attempt to try the alternating series test, which tells us if we have a series in the form

`sum_(n=1)^oo(-1)^nb_n` and `lim_(n->oo)b_n = 0, b_n<=b_(n+1),` the series converges.

Here,

`b_n=n/(n=1), lim_(n->oo)n/(n+1) = 1 ne 0`

So, the alternating series test is inconclusive.

Instead, we'll use the Divergence Test, and take

`lim_(n->oo)(-1)^n * n/(n+1)`

This limit does not truly exist due to the `(-1)^n`, but since `n/(n+1)` is always getting closer to one, we convince ourselves the terms are alternating signs and approaching one, and so the series diverges.