Answer
See below.
Work Step by Step
$\sum_{i=1}^{4} (-1)^{k+1}\frac{k+1}{k^2}=(-1)^{1+1}\frac{1+1}{1^2}+(-1)^{2+1}\frac{2+1}{2^2}+(-1)^{3+1}\frac{3+1}{3^2}=(1)2+(-1)0.75+(1)\frac{4}{9}=\frac{72}{36}-\frac{27}{36}+\frac{16}{36}=\frac{61}{36}$
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