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

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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))/(((-2)^n)/n^2)|`

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

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

`=2`

Since 2>1, the series diverges.

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

`-2,-1,-1.bar(8),-.bar(8),-2.17,-.39,-3,.99,-5.32,4.92,-12,16.43,...`

Sources:

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