Question

How do you use a power series to find the exact value of the sum of the series `1-(pi/4)^2/(2!)+(pi/4)^4/(4!) -(pi/4)^6/(6!) + …` ?

Answer 1

Since the Maclaurin series
`1-x^2/{2!}+x^4/{4!}-x^6/{6!}+cdots=cosx`,
`1-(pi/4)^2/{2!}+(pi/4)^4/{4!}-(pi/4)^6/{6!}+cdots=cos(pi/4)=1/sqrt{2}`.