By the alternating series test, the series k=15(1)k+1k(k+10)\displaystyle {\sum_{{{k}={1}}}^{\infty}}\frac{{{5}{\left(-{1}\right)}^{{{k}+{1}}}}}{{{k}{\left({k}+{10}\right)}}} converges. Find its sum.
First find the partial fraction decomposition of 5k(k+10)\displaystyle \frac{{5}}{{{k}{\left({k}+{10}\right)}}}.
5k(k+10)=\displaystyle \frac{{5}}{{{k}{\left({k}+{10}\right)}}}=  
Then find the limit of the partial sums.
k=15(1)k+1k(k+10)=\displaystyle {\sum_{{{k}={1}}}^{\infty}}\frac{{{5}{\left(-{1}\right)}^{{{k}+{1}}}}}{{{k}{\left({k}+{10}\right)}}}=  
Enter your answer for the sum as a reduced fraction.