let-f-x-sin-2x-1-find-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-

Question Number 40260 by maxmathsup by imad last updated on 17/Jul/18

l e t f (x) = s i n (2 x)

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 .

Commented by maxmathsup by imad last updated on 18/Jul/18

1) w e h a v e f (x) = 2 s i n (x) c o s (x) \Rightarrow f^{(n)} (x) = 2 \sum_{k = 0}^{n} C_{n}^{k} {(s i n x)}^{(k)} {(c o s x)}^{(n - k)}

= 2 \sum_{k = 0}^{n} C_{n}^{k} s i n (x + \frac{k π}{2}) c o s (x + \frac{(n - k) π}{2}) \Rightarrow

f^{(n)} (0) = 2 \sum_{k = 0}^{n} C_{n}^{k} s i n (\frac{k π}{2}) c o s (\frac{(n - k) π}{2})

= 2 \sum_{p = 0}^{[\frac{n - 1}{2}]} C_{n}^{2 p + 1} s i n (\frac{(2 p + 1) π}{2}) c o s (\frac{(n - 2 p - 1) π}{2})

= 2 \sum_{p = 0}^{[\frac{n - 1}{2}]} {(- 1)}^{p} C_{n}^{2 p + 1} c o s (\frac{(n - 1) π}{2} - p π)

= 2 \sum_{p = 0}^{[\frac{n - 1}{2}]} C_{n}^{2 p + 1} c o s (\frac{(n - 1) π}{2})

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

f (x) = \sum_{n = 1}^{\infty} \frac{2}{n!} c o s (\frac{(n - 1) π}{2}) (\sum_{p = 0}^{[\frac{n - 1}{2}]} C_{n}^{2 p + 1}) x^{n}

Commented by maxmathsup by imad last updated on 18/Jul/18

2) a n o t h e r m e t h o d w e h a v e s i n t = I m (e^{i t}) = I m (\sum_{n = 0}^{\infty} \frac{{(i t)}^{n}}{n!})

= I m (\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} t^{2 n}}{(2 n)!} + i \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} t^{2 n + 1}}{(2 n + 1)!}) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} t^{2 n + 1}}{(2 n + 1)!} \Rightarrow

s i n (2 x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} 2^{2 n + 1} x^{2 n + 1}}{(2 n + 1)!} a n d t h e r a d i u s o f c o n v e r g e n c e i s R = + \infty