Consider the series ∑n=1∞(−1)nn2. Evaluate the the following limit.?

The answer is as follows:

a_(n+1) = (-1)^(n+1)/(n+1)² and an = (-1)ⁿ/n².

Their ratio and simplifying [highschool algebra] gives -1*(n/(n+1))². 
Your limit requires the absolute value of this, hence you simply have  (n/(n+1))².

Whats the limit of that? Or, recall that this can be written as

( 1/(1 + 1/n) )². If n ~ ∞, what does it give? You have your answer!