study-the-convergence-of-n-1-1-n-ln-1-1-n-1-x-

Question Number 36742 by prof Abdo imad last updated on 04/Jun/18

s t u d y t h e c o n v e r g e n c e o f

\sum_{n = 1}^{\infty} {(- 1)}^{n} l n (1 + \frac{1}{n (1 + x)}) .

Commented by maxmathsup by imad last updated on 04/Aug/18

l e t f_{n} (x) = {(- 1)}^{n} l n (1 + \frac{1}{n (1 + x)}) f o r x \neq - 1 w e h a v e (f_{n}) a r e c o n t i n u e s a n d

f_{n} (x) \sim {(- 1)}^{n} \frac{1}{n (1 + x)} a n d \sum_{n ⩾ 1} \frac{{(- 1)}^{n}}{n (1 + x)} = \frac{1}{1 + x} \sum_{n ⩾ 1} \frac{(- 1)}{n} i s a c o n v e g e n t

s e r i e s o 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) i s a s s u r e d