# 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)?

Apr 3, 2017

The series diverges.

#### Explanation:

The series ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} {n}^{2} / {\left(n + 1\right)}^{2}$ is made up of two parts.

The ${\left(- 1\right)}^{n + 1}$ part alternates between $1$ and $- 1$, only changing the sign of each term.

The meat of the sequence is ${n}^{2} / {\left(n + 1\right)}^{2}$. Note that ${\lim}_{n \rightarrow \infty} {n}^{2} / {\left(n + 1\right)}^{2} = 1$.

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