Question

Evaluate:

$\frac{1+\frac{1}{2!}+\frac{1}{3!}+.......}{1+\frac{1}{3!}+\frac{1}{5!}+.......}$.

Answer 1

We've,

$e=1+\frac{1}{1!}+\frac{1}{2!}+.......$.

And,

$e^{-1}=1-\frac{1}{1!}+\frac{1}{2!}-......$.

Then $e+e^{-1}=2(1+\frac{1}{2!}+\frac{1}{4!}+......)$.....(1) and $e-e^{-1}=2(1+\frac{1}{3!}+\frac{1}{5!}+......)$.....(2).

Now,

$\frac{1+\frac{1}{2!}+\frac{1}{3!}+.......}{1+\frac{1}{3!}+\frac{1}{5!}+.......}$

$=\frac{\frac{1}{2}(e+e^{-1})}{\frac{1}{2}(e-e^{-1})}$

$=\frac{e^2+1}{e^2-1}$.