let f(x)= Σ_(n=1) ^∞ ((sin(nx))/n) x^n 1) prove that f is C^1 on ]−1,1[ 2)calculate f^′ (x) and prove that f(x)=arctan( ((xsinx)/(1−x cosx)))


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Question Number 36747 by prof Abdo imad last updated on 05/Jun/18

$l e t f (x) = \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} x^{n}$ $1) p r o v e t h a t f i s C^{1} o n] - 1, 1 [$ $2) c a l c u l a t e f^{^{'}} (x) a n d p r o v e t h a t$ $f (x) = a r c t a n (\frac{x s i n x}{1 - x c o s x})$
Commented by prof Abdo imad last updated on 06/Jun/18

$t h e f u n c t i o n f_{n} (x) = \frac{s i n (n x)}{n} x^{n} a r e C^{1} o n] - 1, 1 [$ $a n d f_{n}^{^{'}} (x) = s i n (n x) x^{n - 1} a r e c o n t i n u e s o n] - 1, 1 [$ $a l s o Σ f_{n}^{^{'}} (x) c o n v e r g e s u n i f . o n] - 1, 1 [$ $Σ f_{n} (x) c o n v . u n i f . f i s C^{1} o n] - 1, 1 [$ $2) w e h a v e f^{^{'}} (x) = \sum_{n = 1}^{\infty} s i n (n x) x^{n - 1}$ $= I m (\sum_{n = 1}^{\infty} e^{i n x} x^{n - 1}) b u t$ $\sum_{n = 1}^{\infty} e^{i n x} x^{n - 1} = \sum_{n = 0}^{\infty} e^{i (n + 1) x} x^{n}$ $= e^{i x} \sum_{n = 0}^{\infty} {(x e^{i x})}^{n} = e^{i x} \frac{1}{1 - {x e}^{i x}}$ $= \frac{1}{e^{- i x} - x} = \frac{1}{c o s x - i s i n x - x}$ $= \frac{c o s x - x + i s i n (x)}{{(c o s x - x)}^{2} + {s i n}^{2} x} \Rightarrow$ $f^{^{'}} (x) = \frac{s i n x}{1 - 2 x c o s x + x^{2}} \Rightarrow$ $f (x) = \int \frac{s i n x}{x^{2} - 2 x c o s x + 1} d x + c . . . . b e c o n t i n u e d . . .$
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