Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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

Find: Σ_(n=1) ^∞ ln (n) = ?

$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{ln}\:\left(\mathrm{n}\right)\:=\:? \\ $$

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

this serie diverges to +∞

$$\mathrm{this}\:\mathrm{serie}\:\mathrm{diverges}\:\mathrm{to}\:+\infty \\ $$

Answered by alephzero last updated on 15/Jan/22

Σ_(n=1) ^∞ ln n = ? ln 1 + ln 2 + ... = ln (1 ∙ 2...) = = ln (Π_(k=1) ^∞ k) ⇒ Σ_(n=1) ^∞ ln n = ln (Π_(k=1) ^∞ k) lim_(x→∞) (Π_(k=1) ^x k) = ∞ lim_(x→∞) (ln x) = ∞ ⇒ Σ_(n=1) ^∞ ln n = ∞

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{ln}\:{n}\:=\:? \\ $$$$\mathrm{ln}\:\mathrm{1}\:+\:\mathrm{ln}\:\mathrm{2}\:+\:...\:=\:\mathrm{ln}\:\left(\mathrm{1}\:\centerdot\:\mathrm{2}...\right)\:= \\ $$$$=\:\mathrm{ln}\:\left(\underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}{k}\right) \\ $$$$\Rightarrow\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{ln}\:{n}\:=\:\mathrm{ln}\:\left(\underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}{k}\right) \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\underset{{k}=\mathrm{1}} {\overset{{x}} {\prod}}{k}\right)\:=\:\infty \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{ln}\:{x}\right)\:=\:\infty \\ $$$$\Rightarrow\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{ln}\:{n}\:=\:\infty \\ $$

Commented by HongKing last updated on 15/Jan/22

thank you so much dear Sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{dear}\:\mathrm{Sir} \\ $$

Commented by alephzero last updated on 16/Jan/22

You are welcome, Sir.

$$\mathrm{You}\:\mathrm{are}\:\mathrm{welcome},\:\mathrm{Sir}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com