I have recently been doing some work that involves long, messy manipulations with lots of geometric series. A typical such series, which would only be one of a number of such terms in a formula, is:(adsbygoogle = window.adsbygoogle || []).push({});

$$

\sum_{t=h+1}^{T-h} \left(1-(1-\theta)^{T-t-h+1}\right)

\\

$$

It's not difficult to simplify this and get rid of the summation, but it takes me about five minutes a go (I'm slow, I know!) and there are dozens of these in the work.

I was wondering whether a symbolic algebra program like Maple might be able to do this for me. I don't have Maple but I've been experimenting with a free software equivalent called Maxima.

As far as I can understand from the Help, Maxima would not be able to process the above formula because, while it does have functions that deal with summing geometric series, they seem to require that the series limits be numeric constants, not variables.

I would be happy to pay for software that did this, if such a thing exists. Maple seems to be the best known. Does anybody know if Maple could sum the above and get the simplified version, which is

$$T-2h-(1-\theta)\frac{1-(1-\theta)^{T-2h}}{\theta}$$

Also, if anybody is familiar with Maxima, can you tell me if I'm correct that Maxima could not do the above?

Are there other programs that could do this?

Supplementary question: some of my geometric series have negative or constant-multiplied exponents, like

$$\sum_{i=0}^{h-1} (1-\theta)^{2(h-i-1)}\left(1-(1-\theta)^{i+1}\right)^2$$

Does anybody know if any of the symbolic algebra programs are clever enough to recognise that this can be decomposed into a series of powers of ##\left((1-\theta)^{-2i}\right)## and ##\left((1-\theta)^{-i}\right)## and then use the formula for the sum of a geometric series to simplify them?

Thank you for any help and suggestions.

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# Variable summation limits in Maple or similar programs

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