study-the-covergence-of-u-n-with-u-n-n-n-n-1-1-

Question Number 31984 by abdo imad last updated on 17/Mar/18

s t u d y t h e c o v e r g e n c e o f Σ u_{n} w i t h

u_{n} =^{n} \sqrt{\frac{n}{n + 1}} - 1 .

Commented by prof Abdo imad last updated on 22/Mar/18

w e h a v e u_{n} = {(\frac{n}{n + 1})}^{\frac{1}{n}} - 1

= {(1 - \frac{1}{n + 1})}^{\frac{1}{n}} - 1 b u t w e h a v e

{(1 + x)}^{α} \sim 1 + α x f o r x \in V (0) \Rightarrow

{(1 - \frac{1}{n + 1})}^{\frac{1}{n}} \sim 1 - \frac{1}{n (n + 1)} \Rightarrow

u_{n} \sim - \frac{1}{n (n + 1)} \sim - \frac{1}{n^{2}} d u e t o c o n v e r g e n c e o f

Σ \frac{1}{n^{2}} t h e s e r i e Σ u_{n} c o n v e r g e s .