Strategy for Series — Question 1

*Question 1 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{n}{3^n}$.

1. Choose an efficient convergence test and justify the choice.

2. Determine whether the series converges.

3. If possible, use a known power-series identity to find its exact sum.

*Question 1 – Solution

For $a_n=n/3^n$, $$\frac{a_{n+1}}{a_n}=\frac{n+1}{3n}\longrightarrow\frac13<1.$$ The Ratio Test is quick and proves convergence.

Begin with $$\sum_{n=0}^{\infty}x^n=\frac1{1-x},\qquad |x|<1.$$ Differentiate and multiply by $x$: $$\sum_{n=1}^{\infty}nx^n=\frac{x}{(1-x)^2}.$$ At $x=1/3$, $$\sum_{n=1}^{\infty}\frac{n}{3^n}=\frac{1/3}{(1-1/3)^2}=\boxed{\frac34}.$$

The Ratio Test classifies the series; generating-function recognition gives both convergence and the exact value.