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Question Number 175740 by daus last updated on 06/Sep/22

Answered by Ar Brandon last updated on 06/Sep/22

$$S\left(t\right)=\underset{n=1}{\sum ^{\mathrm{\infty }}}{t}^n=\frac{t}{1-t},\mid t\mid <1$$$$\Rightarrow S{}^{\prime }\left(t\right)=\underset{n=1}{\sum ^{\mathrm{\infty }}}{nt}^{n-1}=\frac{1}{{(1-t)}^2}$$$$\Rightarrow \underset{n=1}{\sum ^{\mathrm{\infty }}}{nt}^n=\frac{t}{{(1-t)}^2},t=\left(\frac{1}{5}\right)$$$$\Rightarrow \underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{n}{5^n}=\frac{\frac{1}{5}}{{(1-\frac{1}{5})}^2}=\frac{1}{5}\times {\left(\frac{5}{4}\right)}^2=\frac{5}{16}$$

Answered by Ar Brandon last updated on 06/Sep/22

$$S=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}+\frac{5}{5^5}+\frac{6}{5^6}+\cdot \cdot \cdot$$$$\frac{1}{5}S=\frac{1}{5^2}+\frac{2}{5^3}+\frac{3}{5^4}+\frac{4}{5^5}+\frac{5}{5^6}+\frac{6}{5^7}+\cdot \cdot \cdot$$$$S-\frac{1}{5}S=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}+\cdot \cdot \cdot$$$$\frac{4}{5}S=\frac{\left(\frac{1}{5}\right)}{1-\frac{1}{5}}=\frac{1}{5}\times \frac{5}{4}=\frac{1}{4}$$$$\Rightarrow S=\frac{5}{16}$$

Commented by daus last updated on 07/Sep/22

$$howtogetaRHSofthethirdline?$$

Commented by Ar Brandon last updated on 07/Sep/22

$$\mathrm{You}\mathrm{do}\mathrm{the}\mathrm{subtraction}\mathrm{using}\mathrm{the}\mathrm{values}$$$$\mathrm{from}\mathrm{the}2\mathrm{previous}\mathrm{lines}.\mathrm{That}\mathrm{is},$$$$\frac{1}{5}+(\frac{2}{5^2}-\frac{1}{5^2})+(\frac{3}{5^3}-\frac{2}{5^3})+(\frac{4}{5^4}-\frac{3}{5^4})+\cdot \cdot \cdot$$

Commented by peter frank last updated on 06/Sep/22

$$\mathrm{thanks}$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

Answered by Rasheed.Sindhi last updated on 06/Sep/22

$$S=\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}+\frac{5}{5^5}+\frac{6}{5^6}+\cdot \cdot \cdot$$$$5S=\frac{1}{1}+\frac{2}{5}+\frac{3}{5^2}+\frac{4}{5^3}+\frac{5}{5^4}+\frac{6}{5^5}+\cdot \cdot \cdot$$$$5S-S=\frac{1}{1}+\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...$$$$4S=\frac{1}{1-\frac{1}{5}}=\frac{5}{4}$$$$S=\frac{5}{16}$$

Commented by peter frank last updated on 06/Sep/22

$$\mathrm{thanks}$$

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