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

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2 Answers
Mar 28, 2018

neither

Explanation:

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

diverges by nth term test

Mar 28, 2018

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.