let-f-n-x-n-x-e-nx-with-x-gt-0-1-study-the-simple-and-uniform-convervence-for-f-n-x-2-let-S-x-n-1-f-n-x-prove-that-S-x-1-x-x-0-

Question Number 33698 by math khazana by abdo last updated on 22/Apr/18

l e t f_{n} (x) = n^{x} e^{- n x} w i t h x > 0

1) s t u d y t h e s i m p l e a n d u n i f o r m c o n v e r v e n c e f o r

Σ f_{n} (x)

2) l e t S (x) = \sum_{n = 1}^{\infty} f_{n} (x) . p r o v e t h a t

S (x) \sim \frac{1}{x} (x \to 0^{+})