Question

Find the sum of this series by converting it into a geometric one.

Answer 1

\[\sum_{1}^{\inf}(\frac{ (2^{n} + 6^{n}) }{ 9^n }\]

Answer 2

@AccessDenied

Answer 3

It seems you could just distribute the denominator to both 2^n and 6^n, then deal with the two as separate geometric series since: a^n / b^n = (a / b)^n

Answer 4

I am stuck with after distributing the denominator, how do you convert it to a geometric series?

Answer 5

What is the general definition for your geometric series?

Answer 6

ar^(n-1)? where a is the first number and r is the ratio correct?

Answer 7

Yep. And we want to make sure that |r| < 1 so that the series converges. When you distribute, you have:
\( \displaystyle \sum_{n=1}^{\infty} \dfrac{2^n}{9^n} + \dfrac{6^n}{9^n} \)
The numerator and denominator are raised to the same power, so you can rewrite like this:
\( \dfrac{a^n}{b^n} = \left( \dfrac{a}{b} \right)^n \)
That seems familiar? :)

Answer 8

Yes thats right. So this is now a geometric series?

Answer 9

at a = 1?

Answer 10

Yes. We separate the sum into two individual series, each has a=1 and the common ratio is that whole fraction raised to the power of n.

Answer 11

okay thanks a lot! I really appreciate your time!

Answer 12

Glad to help! :)