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Evaluate the sum \[\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots \]

Guest Oct 2, 2017
 #1
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1/3 + 2/3^2 + 3/3^3 + 4/3^4 + 5/3^5 +............+ k/3^k =3/4

OK, CPhill or hectictar: use your brilliant algebraic knowledge to prove this. I can't !!.

Guest Oct 2, 2017
 #2
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I don't know how to prove this.....but maybe hectictar does......if not.....hang on for heureka........he's usually pretty good at these kind of things  !!!!

 

[ I always learn something from his answers ]

 

 

cool cool cool

CPhill  Oct 2, 2017
 #3
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Don't wait up for me!! I never really learned how to do these "sum" things!!

hectictar  Oct 2, 2017
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Consider the infinite Geometric series \(\displaystyle \frac{1}{x}+\frac{1}{x^{2}}+\frac{1}{x^{3}}+\dots\)  .

Its sum to infinity (I think that there is a mistake in the question, there should be a plus sign and a series of dots after that last term), is \(\displaystyle \frac{1}{x-1}\).

Equate the two, differentiate both sides, and then substitute x = 3.

 

Tiggsy

Bertie  Oct 2, 2017
edited by Guest  Oct 2, 2017

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