$In$ this problem, consider the periodic function $f$ defined $by$

hat$(f{)}_{n}f(x),S_5(f,x)S_{11}(f,x)f(x),{}_5(f,x){}_{11}(f,x)x=\frac{}{2}1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-$dotsx$=\frac{}{4}1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{13}-\frac{1}{15}+$dotsx$=\frac{}{3}1-\frac{1}{5}+\frac{1}{7}-\frac{1}{11}+\frac{1}{13}-\frac{1}{17}+\frac{1}{19}-$dotsx$=\frac{}{6}\frac{2}{3}1+\frac{1}{5}-\frac{1}{7}-\frac{1}{11}+\frac{1}{13}+\frac{1}{17}-\frac{1}{19}-\frac{1}{23}+$dots$1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\frac{1}{9^2}+$dots

Please make sure $to$ show all the steps please!