Chapter 8.6, Problem 60E

Alternating p-series Given that ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{1}{k^4}}=\frac{{\pi }^4}{90},$show that ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{{(-1)}^{k+1}}{k^4}}=\frac{7{\pi }^4}{720}.$(Assume the result of Exercise 63.)