let-S-x-n-0-f-n-x-with-f-n-x-1-n-n-x-n-x-0-1-prove-that-S-id-defined-calculate-S-1-and-prove-that-x-gt-0-xS-x-S-x-1-1-e-2-prove-that-S-is-C-on-R-3

Question Number 33699 by math khazana by abdo last updated on 22/Apr/18

l e t S (x) = \sum_{n = 0}^{\infty} f_{n} (x) w i t h f_{n} (x) = \frac{{(- 1)}^{n}}{n! (x + n)}

x \in] 0, + \infty [

1) p r o v e t h a t S i d d e f i n e d . c a l c u l a t e S (1) a n d

p r o v e t h a t \forall x > 0 x S (x) - S (x + 1) = \frac{1}{e}

2) p r o v e t h a t S i s C^{\infty} o n R^{+ *}

3) p r o v e t h a t S (x) \sim \frac{1}{x} (x \to 0^{+}) .

Commented by math khazana by abdo last updated on 29/Apr/18

w e h a v e ∣ f_{n} (x) ∣= \frac{1}{n! (x + n)} ⩽ \frac{1}{n (n!)} b u t t h e s e S r i e

Σ \frac{1}{n (n!)} i s c o n v e r g e n t s o S i s d e f i n e d o n] 0, + \infty [

S (1) = \sum_{n = 0}^{\infty} f_{n} (1) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (n + 1)}

l e t w (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (n + 1)} x^{n + 1}

w^{^{'}} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{n}}{n!} = \sum_{n = 0}^{\infty} \frac{{(- x)}^{n}}{n!} = e^{- x} \Rightarrow

w (x) = - e^{- x} + λ w e h a v e w (0) = 0 = - 1 + λ \Rightarrow

λ = 1 \Rightarrow w (x) = 1 - e^{- x}

S (1) = w (1) = 1 - \frac{1}{e} .

x S (x) - S (x + 1) = \sum_{n = 0}^{\infty} \frac{x {(- 1)}^{n}}{n! (x + n)} - \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (x + n + 1)}

= 1 + \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n!} (\frac{x + n - n}{x + n}) - \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (x + n + 1)}

= \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n!} - \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{(n - 1)! (x + n)} - \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (x + n + 1)}

= \frac{1}{e} - \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n + 1}}{n! (x + n + 1)} - \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n! (x + n + 1)}

= \frac{1}{e} \Rightarrow x S (x) - S (x + 1) = \frac{1}{e}

Commented by math khazana by abdo last updated on 29/Apr/18

2) d u e t o u n i f o r m c o n v e r g e n c e a n d f_{n} a r e C^{\infty}

S w i l l b e C^{} o n] 0, + \infty [a n d

S^{(p)} (x) = \sum_{n = 0}^{\infty} f_{n}^{(p)} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n!} \frac{{(- 1)}^{p} p!}{{(x + n)}^{p + 1}} .

Commented by math khazana by abdo last updated on 29/Apr/18

3) w e h a v e p r o v e d t h a t x S (x) - S (x + 1) = \frac{1}{e} \Rightarrow

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

x S (x) - 1 + \frac{1}{e} \sim \frac{1}{e} \Rightarrow S (x) \sim \frac{1}{x} (x \to 0^{+}) .