Question

Does this sequence converge or diverge?

Does the sequence `a_n=(-1)^n *n+n` converge or diverge? If the sequence does converge, what number do the terms of the sequence converge to?

My answer is that it converges to 0. Am I correct?

Answer 1

This series diverges, since the even-valued terms in `n` get increasingly large without limit, and the odd-valued terms all equal zero.

Explanation:

If you write the first two terms:

`a_1 = (-1)^1 *n +n = -n+n=0`

`a_2 = (-1)^2 *n + n= n+n = 2n`

you quickly see that this process repeats such that for every `n` that is odd, the term is zero and for every `n` that is even, the term is `2n`.

As `n` gets steadily larger, the even terms grow as an infinite sum of non-zero values that get increasingly large, while the odd terms vanish.

Therefore, the series diverges.