Question

If $f\left(x\right)={\sum }_{n=1}\frac{\sin nx}{4^n}$ and ${\int }_0^{\pi }f\left(x\right)dx=\log \left(\frac{m}{n}\right)$, then the value of $(m+n)$ is

Answer 1

$f=\frac{\sin x}{4}+\frac{\sin 2x}{4^2}+\frac{\sin 3x}{4^3}+...........$

${\int }_0^{\pi }f={[\frac{-\cos x}{4}-\frac{\cos 2x}{{2.4}^2}-\frac{\sin 3x}{{3.4}^3}]}_0^{\pi }$

$I=-(\frac{-1-1}{4}+\frac{(1-1)}{2-4^2}+\frac{-1-1}{{3.4}^3}+.....)$

$I=2(\frac{1}{4}+\frac{1}{{3.4}^3}+\frac{1}{{5.4}^5}+.........)$ _____ (1)

Now,

$\log (1+x)=\frac{x}{1}-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+......$ _________(2)

$\log (1-x)=-\frac{x}{1}-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}$ ________ (3)

Equation $\left(2\right)$ - equation $\left(3\right)$

$\log \left(\frac{1+x}{1-x}\right)=2(\frac{x}{1}+\frac{x^3}{3}+\frac{x^5}{5}+.....)$

Comparing this with equation $\left(1\right)$ $\Rightarrow$

$\therefore I=\log \left(\frac{1+1/4}{1-1/4}\right)=\log \left(\frac{5}{3}\right)$

$\therefore m+n=5+3=8$