Find-n-1-ln-n-

Question Number 164240 by HongKing last updated on 15/Jan/22

Find : \underset{n = 1}{\sum^{\infty}} \ln (n) = ?

Answered by mathmax by abdo last updated on 15/Jan/22

this serie diverges to + \infty

Answered by alephzero last updated on 15/Jan/22

\underset{n = 1}{\sum^{\infty}} \ln n = ?

\ln 1 + \ln 2 + \dots = \ln (1 \cdot 2 \dots) =

= \ln (\underset{k = 1}{\prod^{\infty}} k)

\Rightarrow \underset{n = 1}{\sum^{\infty}} \ln n = \ln (\underset{k = 1}{\prod^{\infty}} k)

\lim_{x \to \infty} (\underset{k = 1}{\prod^{x}} k) = \infty

\lim_{x \to \infty} (\ln x) = \infty

\Rightarrow \underset{n = 1}{\sum^{\infty}} \ln n = \infty

Commented by HongKing last updated on 15/Jan/22

thank you so much dear Sir

Commented by alephzero last updated on 16/Jan/22

You are welcome, Sir .