Question Number 174500 by Mastermind last updated on 02/Aug/22
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{infinite} \\ $$$$\mathrm{sum}: \\ $$$$\mathrm{1}+\frac{\mathrm{3}}{\pi}+\frac{\mathrm{3}}{\pi^{\mathrm{2}} }+\frac{\mathrm{3}}{\pi^{\mathrm{3}} }+\frac{\mathrm{3}}{\pi^{\mathrm{4}} }+\frac{\mathrm{3}}{\pi^{\mathrm{5}} }+... \\ $$$$ \\ $$$$\mathrm{Mastermind} \\ $$
Answered by Lordose last updated on 02/Aug/22
$$\mathrm{I}\:=\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\boldsymbol{\pi}}\:+\:\frac{\mathrm{3}}{\boldsymbol{\pi}^{\mathrm{2}} }\:+\:\frac{\mathrm{3}}{\boldsymbol{\pi}^{\mathrm{3}} }\:+\:\centerdot\:\centerdot\:\centerdot \\ $$$$\mathrm{I}\:=\:\mathrm{3}\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\boldsymbol{\pi}^{\mathrm{k}} }\:−\:\mathrm{2} \\ $$$$\mathrm{I}\:=\:\mathrm{3}\left(\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{\boldsymbol{\pi}}}\right)\:−\:\mathrm{2}\:=\:\frac{\mathrm{3}\boldsymbol{\pi}}{\boldsymbol{\pi}−\mathrm{1}}\:−\:\mathrm{2}\:\:\:\blacktriangle\blacktriangle\blacktriangle \\ $$
Commented by Mastermind last updated on 02/Aug/22
$$\mathrm{Thanks},\:\mathrm{but}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{meaning}\:\mathrm{of} \\ $$$$\mathrm{triangle}\:\mathrm{symbol}? \\ $$