Answer
$\dfrac{7\pi^4}{720} $
Work Step by Step
We have $1+\dfrac{1}{2^4}+\dfrac{1}{3^4}+\dfrac{1}{4^4}+......=\dfrac{\pi^4}{90} ~~~~~(a)$
Let us consider that $S=1-\dfrac{1}{2^4}+\dfrac{1}{3^4}-\dfrac{1}{4^4}+......~~~~(b)$
Next, we will subtract equation (b) from equation (a) to obtain:
$\dfrac{\pi^4}{90}-S=(1+\dfrac{1}{2^4}+\dfrac{1}{3^4}+\dfrac{1}{4^4}+......)-(1-\dfrac{1}{2^4}+\dfrac{1}{3^4}-\dfrac{1}{4^4}+......)$
or, $=2 [\dfrac{1}{2^4}+\dfrac{1}{4^4}+....]$
or, $=2 \Sigma_{k=1}^{\infty} \dfrac{1}{(2k)^4}$
or, $=\dfrac{1}{8} \times \dfrac{\pi^4}{90} $
Thus, $S=\dfrac{\pi^4}{90} -\dfrac{\pi^4}{720} =\dfrac{7\pi^4}{720} $
