(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the convergence of this seriesusing the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is:

The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞

The question is also attached as a .png file

2. Relevant equations

The geometric series with a * r^n is known to:

The p-series (1/n^p) is known to:

- converge if the absolute value of r is smaller than 1
- diverge if the absolute value of r is greater or equal to 1

Please refer to this website for the definition of the Comparison Test and the Limiting Comparison Test:

- converge if p is greater than 1
- divergent if otherwise

3. The attempt at a solution

This is as far as I got:

∑ [(n+1)(3^n) / 2^(2n)] can be split into two series,

∑ [n(3^n) / (4^n)] + ∑ [(3^n) / (4^n)]

The latter series is known to converge because it is a geometric series with r = 3/4

However, I am stuck trying to solve the first series.

Using the Ratio Test to determine whether the series converges or diverges is quite simple, and I have worked it out.

By the way, could anyone tell me how to make the formulas look more ... natural? Instead of using ^ and / as well as a ton of brackets.

Any help is appreciated!

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# Homework Help: Proving the convergence of series

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