Solve Series Convergence Problem: Find Sum

[Yae Miteo]

Homework Statement

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

Homework Equations

$$\sum_{n=1}^{\infty} \frac{1 + 2^n}{3^n}$$

The Attempt at a Solution

Hello,

I have tried to find the sum of this series using both the limit and integral tests, but I cannot get the right answer. My answer is 1/3, but the book says it is 5/2. How can I solve this?

 

[STEMucator]

Homework Statement

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

Homework Equations

$$\sum_{n=1}^{\infty} \frac{1 + 2^n}{3^n}$$

The Attempt at a Solution

Hello,

I have tried to find the sum of this series using both the limit and integral tests, but I cannot get the right answer. My answer is 1/3, but the book says it is 5/2. How can I solve this?

$\sum_{n=1}^{\infty} \frac{1 + 2^n}{3^n}$
$= \sum_{n=1}^{\infty} \frac{1}{3^n} + \sum_{n=1}^{\infty} \frac{2^n}{3^n}$

Can you see the solution now?