Does this sequence converge or diverge?

Question

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?

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Answer 1 · 2017-12-16T21:29:10

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

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.