let f(x)=arctan(1+x) and f_n (x)= e^(−nx) 1) define fof_n (x) and f_n of(x) 2) study the nature of the series Σ_(n=0) ^(+∞) fof_n (x) and Σ_(n=0) ^∞ f


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Question Number 35766 by abdo mathsup 649 cc last updated on 23/May/18

$l e t f (x) = a r c t a n (1 + x) a n d f_{n} (x) = e^{- n x}$ $1) d e f i n e {f o f}_{n} (x) a n d f_{n} o f (x)$ $2) s t u d y t h e n a t u r e o f t h e s e r i e s \sum_{n = 0}^{+ \infty} {f o f}_{n} (x)$ $a n d \sum_{n = 0}^{\infty} f_{n} o f (x) .$
Commented by maxmathsup by imad last updated on 02/Sep/18

$1) w e h a v e {f o f}_{n} (x) = f (f_{n} (x)) = a r c t a n (1 + f_{n} (x)) = a r c t a n (1 + e^{- n x})$ $f_{n} 0 f (x) = f_{n} (f (x)) = e^{- n f (x)} = e^{- n a r c t a n (1 + x)}$ $2) \sum_{n = 0}^{\infty} {f o f}_{n} (x) = \sum_{n = 0}^{\infty} a r c t a n (1 + e^{- n x}) = Σ u_{n} (x)$ $i f x < 0 {l i m}_{n \to + \infty} a r c t a n (1 + e^{- n x}) \neq 0 s o Σ u_{n} (x) d i v e r g e s$ $i f x > 0 u_{n} (x) \sim e^{- n x} (n \to + \infty) b u t Σ e^{- n x} c o n v e r g e s s o t h e c o n v e r g e n c e$ $s i m p l e o f Σ u_{n} (x) i s a s s u r e d . a l s o$ $Σ f_{n} o f = \sum_{n} e^{- n a r c t a n (1 + x)} = \sum_{n} {(e^{- a r c t a n (1 + x)})}^{n} \Rightarrow$ $i f a r c t a n (1 + x) > 0 t h e s e r i e c o n v e r g e s$ $i f a r c t a n (1 + x) < 0 t h e s e r i e d i v e r g e s .$
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