Question Number 86598 by Rio Michael last updated on 29/Mar/20
$$\:\mathrm{write}\:\mathrm{out}\:\mathrm{the}\:\mathrm{general}\:\mathrm{summation}\:\mathrm{formula}\:\mathrm{for} \\ $$$$\:\mathrm{the}\:\mathrm{maclaurin}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\:\:\frac{\mathrm{1}}{\mathrm{2}}\:\left(\mathrm{cos}\:{x}\:+\:\mathrm{cosh}\:{x}\right) \\ $$
Commented by mathmax by abdo last updated on 31/Mar/20
$${cosx}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}{x}^{\mathrm{2}{n}} \:\:\:{and}\:{ch}\left({x}\right)=\frac{{e}^{{x}} +{e}^{−{x}} }{\mathrm{2}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{x}^{{n}} }{{n}!}\:+\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}{x}^{{n}} \right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{{n}!}\left\{\mathrm{1}+\left(−\mathrm{1}\right)^{{n}} \right\}{x}^{{n}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}{n}} \:\Rightarrow}{\left(\mathrm{2}{n}\right)!} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left({cosx}\:+{ch}\left({x}\right)\right)\:=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}{x}^{\mathrm{2}{n}} \:+\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}\right)!}{x}^{\mathrm{2}{n}} \right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sum_{{n}=\mathrm{0}} ^{\infty} \frac{\mathrm{1}+\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}{x}^{\mathrm{2}{n}} \:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{4}{n}} }{\left(\mathrm{4}{n}\right)!} \\ $$