By the alternating series test, the series $\displaystyle {\sum_{{{k}={1}}}^{\infty}}\frac{{{2}{\left(-{1}\right)}^{{{k}+{1}}}}}{{{k}{\left({k}+{4}\right)}}}$ converges. Find its sum.
First find the partial fraction decomposition of $\displaystyle \frac{{2}}{{{k}{\left({k}+{4}\right)}}}$.
$\displaystyle \frac{{2}}{{{k}{\left({k}+{4}\right)}}}=$ Preview Question 6 Part 1 of 2  
Then find the limit of the partial sums.
$\displaystyle {\sum_{{{k}={1}}}^{\infty}}\frac{{{2}{\left(-{1}\right)}^{{{k}+{1}}}}}{{{k}{\left({k}+{4}\right)}}}=$ Preview Question 6 Part 2 of 2  
Enter your answer for the sum as a reduced fraction.