How to calculate this series

1. Jun 6, 2005

phonic

Is it possible to get a analytical result for this series? It looks simple:

$\sum_{k=1} ^t a^{t-k}b^{k-1}$

Thanks a lot!

2. Jun 6, 2005

dextercioby

Hmm.Use [ tex ] & [ /tex ] commands (without the spaces) for opening & closing tex tags.

That's no series,it's a polynomial in 2 variables.

Daniel.

3. Jun 6, 2005

phonic

Thanks for your corection. It's my first time to post message here.

Yes, this is a polynomial, with all coeficient as 1. Is there some method to deal with it?

4. Jun 6, 2005

dextercioby

Yes,try to write some terms in the sum and then see whether you recognize something familiar.

Daniel.

5. Jun 6, 2005

phonic

Thanks for your hints. I think it can be calculated in this way:
$$\sum_{k=1} ^t a^{t-k}b^{k-1} = \frac{a^t}{b}\sum_{k=1} ^t (\frac{b}{a})^{k}$$

6. Jun 6, 2005

dextercioby

It's easier this way:

$$\sum_{k=1}^{t} a^{t-k}b^{k-1}=a^{t-1}b^{0}+a^{t-2}b^{1}+...+a^{1}b^{t-2}+a^{0}b^{t-1}=\frac{a^{t}-b^{t}}{a-b}$$

with "t" uneven.

Daniel.