write-out-the-general-summation-formula-for-the-maclaurin-series-expansion-for-1-2-cos-x-cosh-x-

Question Number 86598 by Rio Michael last updated on 29/Mar/20

write out the general summation formula for

the maclaurin series expansion for \frac{1}{2} (\cos x + \cosh x)

Commented by mathmax by abdo last updated on 31/Mar/20

c o s x = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} x^{2 n} a n d c h (x) = \frac{e^{x} + e^{- x}}{2}

= \frac{1}{2} (\sum_{n = 0}^{\infty} \frac{x^{n}}{n!} + \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n!} x^{n})

= \frac{1}{2} \sum_{n = 0}^{\infty} \frac{1}{n!} {1 + {(- 1)}^{n}} x^{n} = \sum_{n = 0}^{\infty} \frac{x^{2 n} \Rightarrow}{(2 n)!}

\frac{1}{2} (c o s x + c h (x)) = \frac{1}{2} {\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n)!} x^{2 n} + \sum_{n = 0}^{\infty} \frac{1}{(2 n)!} x^{2 n}}

= \frac{1}{2} \sum_{n = 0}^{\infty} \frac{1 + {(- 1)}^{n}}{(2 n)!} x^{2 n} = \sum_{n = 0}^{\infty} \frac{x^{4 n}}{(4 n)!}