let-f-x-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-

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 \dots . b e c o n t i n u e d \dots