why any infinitely differentiable function is a power series? mathematically, if f(x) is A infinitely differentiable function then why f(x)=ax^0 +bx^1 +cx^2 +dx^3 +ex^4 +..... for example sin(x)=x−(x^3 /(3!))+(x^5 /(5!))−(x^7 /(7!))+.....up to∞


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Question Number 21270 by Nayon last updated on 18/Sep/17

$why any infinitely differentiable$ $function is a power series ?$ $m a t h e m a t i c a l l y,$ $i f f (x) i s A i n f i n i t e l y d i f f e r e n t i a b l e$ $f u n c t i o n t h e n w h y$ $f (x) = {a x}^{0} + {b x}^{1} + {c x}^{2} + {d x}^{3} + {e x}^{4} + . . . . .$ $f o r e x a m p l e$ $s i n (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + . . . . . u p t o \infty$
Commented by Nayon last updated on 18/Sep/17

$p l s s o m e o n e c l e a r i f y i t . .$
Commented by 1kanika# last updated on 26/Nov/17

$Because by the Taylor series expansion .$
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