<s>[INST] Sums of the series $1 + (x^2) / 3! +( x^4) / 5! +\cdots$ How can I compute sum of the series ; $$1 + \frac{x^2}{3!}+\frac{x^4}{5!}+\frac{x^6}{7!}+\frac{x^8}{9!}+\cdots$$
I tried to divide it to two pieces such that $$f(x) = 1+\frac{x}{2!}+\frac{x^2}{3!}+\frac{x^3}{4!}+\cdots = \frac{e^x-x}{x}$$ and that $$g(x) = \frac{x}{2!}+\frac{x^3}{4!}+\frac{x^5}{6!}+\frac{x^7}{8!}+\cdots $$
so that answer is equal to $f(x)-g(x)$ but I couldn't proceed since I cannot find any expression for $g(x)$
 [/INST] Note that $xf(x)=\frac{1}{2}(e^x-e^{-x})=\sinh x$.
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