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)#?

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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)#?

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Answer 1 · 2017-04-03T22:08:01

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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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

Explanation:

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