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-n-0-fof-n-x-and-n-0-f-n-of-x-

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 .