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Question Number 2783 by 123456 last updated on 27/Nov/15

$$\text{Double subscripts: use braces to clarify}$$$$\mathrm{proof}\mathrm{that}\gamma \mathrm{is}\mathrm{finite}\mathrm{and}\gamma \in (0,1)$$

Commented by Filup last updated on 27/Nov/15

$$\mathrm{I}\mathrm{am}\mathrm{curious}\mathrm{as}\mathrm{to}\mathrm{how}\mathrm{to}\mathrm{solve}\mathrm{these}$$$$\mathrm{kinds}\mathrm{of}\mathrm{questions}.$$

Commented by Filup last updated on 27/Nov/15

$$\mathrm{What}\mathrm{does}:=\mathrm{mean}?$$$$\mathrm{Same}\mathrm{as}\equiv ?$$

Commented by 123456 last updated on 27/Nov/15

$$:=\mathrm{mean}\mathrm{defined}$$$$\mathrm{ex}:$$$$f\left(x\right):=x$$$$f\left(1\right)=1$$

Commented by Filup last updated on 27/Nov/15

$$Ah\mathrm{I}\mathrm{see}!$$

Answered by prakash jain last updated on 27/Nov/15

$${\gamma }_n={\mathrm{H}}_n-\ln n$$$${\mathrm{H}}_n=1+\frac{1}{2}+\frac{1}{3}+..+\frac{1}{n}=\underset{i=1}{\sum ^n}{a}_i$$$$a_i=\frac{1}{i}$$$$f\left(x\right)=\frac{1}{x}\mathrm{so}\mathrm{that}f\left(i\right)=a_i$$$$\mathrm{Since}f\left(x\right)\mathrm{is}\mathrm{strictly}\mathrm{decreasing}+\mathrm{ve}\mathrm{function}.$$$$\mathrm{From}\mathrm{integral}\mathrm{test}\mathrm{for}\mathrm{series}$$$${\int }_N^{M+1}f\left(x\right)dx⩽\underset{n=N}{\sum ^M}f\left(n\right)⩽f\left(N\right)+{\int }_N^{M}f\left(x\right)dx...\left(A\right)$$$${\mathrm{H}}_{n-1}⩾{\int }_1^n\frac{1}{x}dx=\ln n$$$$S\mathrm{o}{\gamma }_n={\mathrm{H}}_n-\ln n=\frac{1}{n}+{\mathrm{H}}_{n-1}-\ln n>0...\left(1\right)$$$${\gamma }_{n+1}={\gamma }_n+\frac{1}{n+1}-\ln (n+1)+\ln n$$$$\frac{1}{n+1}⩽\ln \frac{n+1}{n}\left(comparingareaofrectangles\right)$$$$\Rightarrow {\gamma }_{n+1}={\gamma }_n-[\ln \frac{n+1}{n}-\frac{1}{n+1}]<{\gamma }_n$$$${\gamma }_n>0and{\gamma }_{n+1}<{\gamma }_n$$$$\mathrm{So}\underset{n\to \mathrm{\infty }}{\lim }{\gamma }_{n}exists\mathrm{and}>0.$$$${\gamma }_1=1-\ln 1=1$$$$\because {\gamma }_{n+1}<{\gamma }_n$$$$0<\gamma <1$$

Commented by RasheedAhmad last updated on 29/Nov/15

$$Whatis{\mathrm{H}}_n?$$

Commented by 123456 last updated on 29/Nov/15

$$\mathrm{harmonic}\mathrm{numbers}$$

Commented by Rasheed Soomro last updated on 29/Nov/15

$$\mathcal{THANKS}!$$

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