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 Evaluate the series 1/3^1 + 4/3^2 + 1/3^3 + 4/3^4 + ...

 May 9, 2022
 #1
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The sum is equal to 5/3.

 May 9, 2022
 #2
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sumfor(n, 1, 100,  1/(3^(2*n - 1))  +  4 / 3^(2*n))==It converges to 7/8

 May 9, 2022
 #3
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Assuming the coefficients are always 1, 4, 1, 4, 1, 4 so on:

 

 

\(\dfrac1{3^1} + \dfrac4{3^2} + \dfrac1{3^3} + \dfrac4{3^4} + \cdots\\ = \dfrac13 \left(1 + \dfrac19 + \dfrac1{9^2} + \cdots\right) + \dfrac4{3^2} \left(1 + \dfrac19 + \dfrac1{9^2}+\cdots\right)\\ = \left(\dfrac{1}{3} + \dfrac49\right)\left(\dfrac1{1 - \dfrac19}\right)\\ = \dfrac78\)

 May 9, 2022

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