Question

How do you use a power series to find the exact value of the sum of the series `1+2+4/(2!) +8/(3!) +16/(4!) + …` ?

Answer 1

Since

`1+x+x^2/{2!}+x^3/{3!}+x^4/{4!}+cdots=e^x`,

by replacing `x` by `2`,

`1+2+2^2/{2!}+2^3/{3!}+2^4/{4!}+cdots=e^2`.

I hope that this was helpful.