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Question Number 104406 by Don08q last updated on 21/Jul/20

$$\mathrm{Evaluate}$$$$\frac{1}{2}+\frac{3}{8}+\frac{3}{16}+\frac{5}{64}+.....$$

Commented by Don08q last updated on 21/Jul/20

$$\frac{1\times 2}{4^1}+\frac{2\times 3}{4^2}+\frac{3\times 4}{4^3}+\frac{4\times 5}{4^4}+.....$$

Answered by JDamian last updated on 21/Jul/20

$$f\left(x\right)\equiv 1+x+x^2+x^3+x^4+\cdot \cdot \cdot =\frac{1}{1-x}\mathrm{\forall }\mid x\mid <1$$$$f^{″}\left(x\right)=1\cdot 2+2\cdot 3\cdot x+3\cdot 4\cdot x^2+4\cdot 5\cdot x^3+\cdot \cdot \cdot =$$$$=\frac{2}{{(1-x)}^3}$$$$f^{″}\left(\frac{1}{4}\right)=2+\frac{2\cdot 3}{4^1}+\frac{3\cdot 4}{4^2}+\frac{4\cdot 5}{4^3}+\frac{5\cdot 6}{4^4}+\cdot \cdot \cdot =$$$$=\frac{2}{{(1-\frac{1}{4})}^3}=\frac{2}{{\left(\frac{3}{4}\right)}^3}=2{\left(\frac{4}{3}\right)}^3$$$$\frac{1}{4}{f}^{″}\left(\frac{1}{4}\right)=\frac{1}{2}+\frac{2\cdot 3}{4^2}+\frac{3\cdot 4}{4^3}+\frac{4\cdot 5}{4^4}+\frac{5\cdot 6}{4^5}+\cdot \cdot \cdot =$$$$=\frac{1}{2}{\left(\frac{4}{3}\right)}^3$$

Commented by Don08q last updated on 22/Jul/20

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}$$

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