Question Number 21270 by Nayon last updated on 18/Sep/17
$$\mathrm{why}\:\mathrm{any}\:\mathrm{infinitely}\:\mathrm{differentiable}\: \\ $$$$\mathrm{function}\:\mathrm{is}\:\mathrm{a}\:\mathrm{power}\:\mathrm{series}? \\ $$$${mathematically}, \\ $$$${if}\:{f}\left({x}\right)\:{is}\:{A}\:{infinitely}\:{differentiable} \\ $$$${function}\:{then}\:{why} \\ $$$${f}\left({x}\right)={ax}^{\mathrm{0}} +{bx}^{\mathrm{1}} +{cx}^{\mathrm{2}} +{dx}^{\mathrm{3}} +{ex}^{\mathrm{4}} +….. \\ $$$${for}\:{example} \\ $$$${sin}\left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}!}+…..{up}\:{to}\infty \\ $$$$ \\ $$
Commented by Nayon last updated on 18/Sep/17
$${pls}\:{some}\:{one}\:{clearify}\:{it}\:.. \\ $$
Commented by 1kanika# last updated on 26/Nov/17
$$\mathrm{Because}\:\mathrm{by}\:\mathrm{the}\:\mathrm{Taylor}\:\mathrm{series}\:\mathrm{expansion}. \\ $$