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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} + \dots . .

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!} + \dots . . 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 .