Is There an Analytical Solution for This Mathematical Series?

[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!

 

[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.

 

[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?

 

[dextercioby]

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

Daniel.

 

[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}

 

[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.