solve if the given series converges or diverges Sum(upper^infinity, lower n=1) (-2)^n/n^2

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embizze eNotes educator| Certified Educator

Determine if `sum_(n=1)^(oo) ((-2)^n)/n^2` converges or diverges.

We apply the ratio test: if `lim_(n->oo)|(a_(n+1))/a_n|<1` then the series converges.


`=lim_(n->oo)|(((-2)^(n+1))/((n+1)^2) * n^2/((-2)^n))|`



Since 2>1, the series diverges.


The finite values of the series oscillate between increasingly negative and positive values: