find-nature-of-n-2-1-n-n-ln-n-1-n-1-

Question Number 52677 by maxmathsup by imad last updated on 11/Jan/19

f i n d n a t u r e o f \sum_{n = 2}^{\infty} {(- 1)}^{n} \sqrt{n} l n (\frac{n + 1}{n - 1}) .

Commented by maxmathsup by imad last updated on 08/Feb/19

S = \sum_{n = 2}^{\infty} {(- 1)}^{n} V_{n} w i t h V_{n} = \sqrt{n} l n (\frac{n + 1}{n - 1}) \Rightarrow V_{n} = \sqrt{n} l n (\frac{n - 1 + 2}{n - 1})

= \sqrt{n} l n (1 + \frac{2}{n - 1}) \sim \sqrt{n} \frac{2}{n - 1} = \frac{\sqrt{n}}{\sqrt{n - 1}} \frac{2}{\sqrt{n - 1}} \sim \frac{2}{\sqrt{n - 1}}

l e t W_{n} = \frac{2}{\sqrt{n - 1}} w e h a v e W_{n} > 0 W_{n} i s d e c r e a s i n g a n d l i m W_{n} = 0 \Rightarrow

Σ {(- 1)}^{n} W_{n} i s a a l t e r n a t e s e r i e c o n v e r g e n t \Rightarrow S i s c o n v e r g e n t .

Answered by tanmay.chaudhury50@gmail.com last updated on 12/Jan/19

\frac{T_{n}}{T_{n - 1}} = \frac{{(- 1)}^{n}}{{(- 1)}^{n - 1}} \times \frac{\sqrt{n}}{\sqrt{n - 1}} \times \frac{l n (\frac{n + 1}{n - 1})}{l n (\frac{n - 1 + 1}{n - 1 - 1})}

= {(- 1)}^{1} \times \frac{1}{\sqrt{1 - \frac{1}{n}}} \times \frac{l n (\frac{1 + \frac{1}{n}}{1 - \frac{1}{n}})}{l n (\frac{1}{1 - \frac{2}{n}})}

= (- 1) \times 1 \times \frac{l n (\frac{1 + 0}{1 - 0})}{l n (\frac{1}{1 - 0})} [s i n c e n \to \infty]

= - 1 \times \frac{l n 1}{l n 1} [h e r e \frac{0}{0} i s c o m i n g]

i f w e c a n c e l l c o m m o n N_{r} a n d D_{r} t h a t i s l n 1

t h e n g i v e n s e r i e s i s c o n v e r g e n t \dots

s o o t h e r s a r e r e q u e s t e d t o C H E C k \dots