How do you determine if the series the converges conditionally, absolutely or diverges given Sigma ((-1)^(n+1)n^2)/(n+1)^2 from [1,oo)?

1 Answer
mason m
Apr 3, 2017

The series diverges.

Explanation:

The series sum_(n=1)^oo(-1)^(n+1)n^2/(n+1)^2 is made up of two parts.

The (-1)^(n+1) part alternates between 1 and -1, only changing the sign of each term.

The meat of the sequence is n^2/(n+1)^2. Note that lim_(nrarroo)n^2/(n+1)^2=1.

So, as the series extends infinitely, alternating terms approaching a value of 1 are continually being added and subtracted, so the series never converges.

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