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Compute \(1 + \frac{3}{5} + \frac{5}{5^2} + \frac{7}{5^3} + \dotsb.\)

 Jul 15, 2023
 #1
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a=sumfor(n, 1, 1000000,  1 + (2*n+1) / 5^n)==It converges to 1 + 7/8 

 Jul 15, 2023
 #2
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S =F / [1 - r]

 

1.875 =1 /[1 - r], solve for r

 

r = 7/15

 

S = 1 / [1 - 7/15] = 1 7/8

 Jul 15, 2023
 #3
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\(\displaystyle S = 1 + \frac{3}{5}+\frac{5}{5^{2}}+\frac{7}{5^{3}}+ \dots \displaystyle = 1+\frac{1+2}{5}+\frac{3+2}{5^{2}}+\frac{5+2}{5^{3}}+\dots \\ \displaystyle = 1 + \frac{1}{5}+\frac{3}{5^{2}}+\frac{5}{5^{3}}+\dots + \frac{2}{5}+\frac{2}{5^{2}}+\frac{2}{5^{3}}+\dots \\ \displaystyle = 1+\frac{1}{5}\left \{1+\frac{3}{5}+\frac{5}{5^{2}}+ \dots \right \} +\frac{2}{5}\left \{1+\frac{1}{5}+\frac{1}{5^{2}}+\frac{1}{5^{3}}+ \dots \right \} \\ \displaystyle =1+\frac{1}{5}S+\frac{2}{5}.\frac{1}{1-(1/5)} \\ \displaystyle \frac{4}{5}S=\frac{3}{2}, \quad S = \frac{15}{8}.\)

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 Jul 16, 2023

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