1. The problem statement, all variables and given/known data Prove the convergence of this series using 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: converge if the absolute value of r is smaller than 1 diverge if the absolute value of r is greater or equal to 1 The p-series (1/n^p) is known to: converge if p is greater than 1 divergent if otherwise Please refer to this website for the definition of the Comparison Test and the Limiting Comparison Test: 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!