Powers problem

$\begin{aligned} B & = 1^{-4} + 3^{-4} + 5^{-4} + 7^{-4} + ... \\ & = 1^{-4} + {\color{#3D99F6} 2^{-4}} + 3^{-4} + {\color{#3D99F6} 4^{-4}} + 5^{-4} + {\color{#3D99F6} 6^{-4}} + 7^{-4} + ... \color{#D61F06} - \left(2^{-4} + 4^{-4} + 6^{-4} + 8^{-4} + ...\right) \\ & = 1^{-4} + 2^{-4} + 3^{-4} + 4^{-4} + ... - 2^{-4} \left(1^{-4} + 2^{-4} + 3^{-4} + 4^{-4} + ...\right) \\ & = A - \frac 1{16} A = \frac {15}{16} \end{aligned}$

$\implies \dfrac AB = \dfrac {16}{15} \implies p.q = \boxed{16.15}$ .

$A=1^{-4}+2^{-4}+3^{-4}+4^{-4}.......$ $B=1^{-4}+3^{-4}+5^{-4}+7^{-4}.......$ Therefore, $A=B+2^{-4}+4^{-4}+6^{-4}+8^{-4}.......$ $A=B+2^{-4}(1^{-4}+2^{-4}+3^{-4}+4^{-4}.......)$ $A=B+2^{-4}(A)$ $A-2^{-4}(A)=B$ $A(1 -2^{-4})=B$ $A(1-1/16)=B$ $A(15/16)=B$ $A/B=16/15$

=16.15 as required