Question

How do you use a power series to find the exact value of the sum of the series `pi-pi^2/2+pi^4/(4!) -pi^6/(6!) + …` ?

Answer 1

Since

`1-x^2/{2!}+{x^4}/{4!}-{x^6}/{6!}+cdots=cosx`,

`1-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots=cos(pi)=-1`

Therefore,

`pi-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots`

`=pi-1+(1-pi^2/{2!}+{pi^4}/{4!}-{pi^6}/{6!}+cdots)`

`=pi-1+(-1)=pi-2`

I hope that this was helpful.