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Question Number 94334 by mathmax by abdo last updated on 18/May/20

l e t f (x) = \frac{s i n x}{x} i f x \neq 0 a n d f (0) = 1

1) {f i n d f}^{(n)} (x) a n d f^{(n)} (0)

2) d e v e l o p p f a t i n t e g r s e r i e s t x_{0} = 0 a n d x_{0} = \frac{π}{2}

Answered by abdomathmax last updated on 18/May/20

1) we have f (x) = \frac{sinx}{x} \Rightarrow f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(\frac{1}{x})}^{(k)} {(sinx)}^{(n - k)}

= \frac{1}{x} \sin (x + \frac{n π}{2}) + \sum_{k = 1}^{n} C_{n}^{k} \frac{{(- 1)}^{k} k!}{x^{k + 1}} \sin (x + \frac{(n - k) π}{2})

2) we have sinx = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1} \Rightarrow

\frac{sinx}{x} = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n}