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Question Number 174500 by Mastermind last updated on 02/Aug/22

$$\mathrm{Find}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{infinite}$$$$\mathrm{sum}:$$$$1+\frac{3}{\pi }+\frac{3}{{\pi }^2}+\frac{3}{{\pi }^3}+\frac{3}{{\pi }^4}+\frac{3}{{\pi }^5}+...$$$$$$$$\mathrm{Mastermind}$$

Answered by Lordose last updated on 02/Aug/22

$$\mathrm{I}=1+\frac{3}{\mathit{\pi }}+\frac{3}{{\mathit{\pi }}^2}+\frac{3}{{\mathit{\pi }}^3}+\cdot \cdot \cdot$$$$\mathrm{I}=3\underset{\mathrm{k}=0}{\sum ^{\mathrm{\infty }}}\frac{1}{{\mathit{\pi }}^{\mathrm{k}}}-2$$$$\mathrm{I}=3\left(\frac{1}{1-\frac{1}{\mathit{\pi }}}\right)-2=\frac{3\mathit{\pi }}{\mathit{\pi }-1}-2▴▴▴$$

Commented by Mastermind last updated on 02/Aug/22

$$\mathrm{Thanks},\mathrm{but}\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{meaning}\mathrm{of}$$$$\mathrm{triangle}\mathrm{symbol}?$$

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