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k-0-2-k-3-k-5-k-




Question Number 150374 by mathdanisur last updated on 11/Aug/21
Σ_(k=0) ^∞  ((2^k  + 3^k )/5^k )  = ?
$$\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{2}^{\boldsymbol{\mathrm{k}}} \:+\:\mathrm{3}^{\boldsymbol{\mathrm{k}}} }{\mathrm{5}^{\boldsymbol{\mathrm{k}}} }\:\:=\:? \\ $$
Answered by eman_64 last updated on 11/Aug/21
   = Σ_(k=0) ^∞ ((2/5))^k +Σ_(k=0) ^∞ ((3/5))^k        = (5/2) +(5/3)         = ((25)/6)
$$\:\:\:=\:\sum_{\mathrm{k}=\mathrm{0}} ^{\infty} \left(\frac{\mathrm{2}}{\mathrm{5}}\right)^{\boldsymbol{\mathrm{k}}} +\sum_{\boldsymbol{\mathrm{k}}=\mathrm{0}} ^{\infty} \left(\frac{\mathrm{3}}{\mathrm{5}}\right)^{\boldsymbol{\mathrm{k}}} \:\: \\ $$$$\:\:\:=\:\frac{\mathrm{5}}{\mathrm{2}}\:+\frac{\mathrm{5}}{\mathrm{3}}\:\:\: \\ $$$$\:\:\:\:=\:\frac{\mathrm{25}}{\mathrm{6}} \\ $$
Commented by mathdanisur last updated on 11/Aug/21
Thank you Ser
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser} \\ $$

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