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**0 **

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**0 **

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 ]

CPhill
Oct 2, 2017

#3

#4**0 **

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