# Thread: Convergence of a Series

1. ## Convergence of a Series

$\displaystyle (2^x+9^x)/18^x$

I have to find the sum of this series from x=1 to infinity, which means that it is either a telescoping or a geometric series. I can't choose which because of its awkward format.
I know that it converges, but I don't what it converges to. Please don't give me the answer. I would just like a hint or a first step of what to do.

$\displaystyle (2^x+9^x)/18^x = (1/9)^x+(1/2)^x$

2. Use:

$\displaystyle \displaystyle\sum_{x=0}^{+\infty}r^x=\dfrac{1}{1-r},\quad (|r|<1)$

3. So it's a geometric series. Should I find the sum of the (1/9)^x and (1/2)^x and just add them together?

4. ^^
Yes.
But becareful, the formula in post#2 is used when the sums starts with x=0.

The actual formula is $\displaystyle S=\dfrac{a}{1-r}$ where a = first term, r = ratio. and |r| must be < 1.

5. Originally Posted by JewelsofHearts
So it's a geometric series. Should I find the sum of the (1/9)^x and (1/2)^x and just add them together?

Use the theorem:

$\displaystyle \sum_{n=1}^{+\infty}u_n,\; \sum_{n=1}^{+\infty}v_n$ both convergent with sums $\displaystyle U,V$ respectively, then $\displaystyle \sum_{n=1}^{+\infty}(u_n+v_n)$ (sum series) is convergent with sum $\displaystyle U+V$ .