Question

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

Answer 1

Since

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

`1+sqrt{2}+{(sqrt{2})^2}/{2!}+{(sqrt{2})^3}/{3!}+{(sqrt{2})^4}/{4!}+cdots=e^{sqrt{2}}`