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Question Number 52679 by maxmathsup by imad last updated on 11/Jan/19

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

1) s t u d y t h e s i m p l e c o n v e r g e n c e o f Σ f_{n} (x)

2) c a l c u l a t e f^{^{'}} (x)

Commented by maxmathsup by imad last updated on 28/Feb/19

1) w e h a v e \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{n + x} = \sum_{n = 0}^{\infty} {(- 1)}^{n} U_{n} (x) w i t h U_{n} (x) = \frac{1}{n + x}

f o r x > 0 f i x e d U_{n} i s d e c r e a s n g a n d {l i m}_{n \to + \infty} U_{n} (x) = 0 s o Σ f_{n} (x) i s

a a l t e r n a t e s e r i e s o t h e s i m p l e c o n v e g e n c e i s a s s u r e d .

2) w e ∣ Σ \frac{{(- 1)}^{n}}{{(n + x)}^{2}} ∣ ⩽ Σ \frac{1}{n^{2}} a n d Σ f_{n}^{^{'}} (x) c o n v e r g e s u n i f . s o i f i s i t s s u m

f^{^{'}} (x) = {(Σ f_{n} (x))}^{^{'}} = Σ f_{n}^{^{'}} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n + 1}}{{(n + x)}^{2}}