find-the-convergence-or-divergence-n-1-n-1-n-

Question Number 30123 by tawa tawa last updated on 16/Feb/18

find the convergence or divergence \underset{n = 1}{\sum^{\infty}} (\frac{n - 1}{n})

Commented by prof Abdo imad last updated on 16/Feb/18

l e t p u t S_{n} = \sum_{k = 1}^{n} \frac{k - 1}{k} w e h a v e

S_{n} = n - \sum_{k = 1}^{n} \frac{1}{k} = n - H_{n} b u t H_{n} = l n (n) + γ + o (1)

\Rightarrow S_{n} = n - l n (n) - γ + o (1) b u t

{l i m}_{n \to \infty} n - l n (n) = {l i m}_{n \to \infty} n (1 - \frac{l n (n)}{n}) = + \infty

{l i m}_{n \to \infty} S_{n} = + \infty a n d f r o m t h a t t h e s e r i e i s

d i v e r g e n t e .

Commented by tawa tawa last updated on 17/Feb/18

God bless you sir .

Commented by abdo imad last updated on 17/Feb/18

m a n y t h a n k s s i r t a w a t a w a \dots