Thread: Series and Sequences help

My book already tells me the answer to a problem and does it step by step, but I have a question b/w a step.

The original question is here:

Sigma from 1 to infinity of 3^(2k)*5^(1-k)

After that, the book says to simplify it to this:

(9^k)/(5^k-1)

After that, this is where I have my problem. It say to simplify it once more to what I have beloew, only this does not make sense to me. How do you break up a 9^k into a 9 by itself? Furthermore, why is a 9 put into the 1/5? Wouldn't it just be 1/5^k-1 times 9^k? I really do not understand simplifying these types of equations so if anyone could help me with this one or link me to a site that shows how to simplify these things, I would greatly appreciate it. The only rule I know about any number ^k is that anything ^k+1 is the same thing as anything times anything ^k. Is there a list of these rules somewhere dealing with numbers to the power of a variable?

9*(9/5)^k-1

edit: and are the defining property of exponents. Letting a=-b you get so . Also, so . You can look the others up on wikipedia I think.

Originally Posted by maddas

Wow, I never would have gotten that. That's too far outside the box for me. Thank you so much.

Originally Posted by Simon777

My book already tells me the answer to a problem and does it step by step, but I have a question b/w a step.

The original question is here:

Sigma from 1 to infinity of 3^(2k)*5^(1-k)

After that, the book says to simplify it to this:

(9^k)/(5^k-1)



... which diverges.

...