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Old 12-08-2005, 08:38 PM
Posts: n/a
Default Re: Infinite multiplication

This math hurts my head. :-| Can one of you tell me what this series would converge to:

M = .98 * .998 * .9998 * .99998 * .999998 * .9999998 * ...

And how do you figure that out?

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I don't know how to find an exact value, but it is a little bit more than .9778. I got that by multiplying the first few terms together, and then convincing myself that the rest of the product doesn't matter very much. There are very few infinite series or products for which the exact value is known, and I don't recognize this particular one as being one of them.

One reason why the rest of the series doesn't matter very much is the following: you are taking the product from n=1 to infty of [1-.2(10)^{-n}]. The log of the n-th term is roughly -.2(10)^{-n} (where by log I mean base e), and this approximation is very accurate for all but the smallest values of n. The logs are thus close to a geometric series, and summing that series from n=k to infty gives (-.2)10^{-k}/.9. Taking k to be 7 or so gives something fairly tiny, so the product of all the terms except the first 6 is essentially 1. Multiplying the first 6 terms of the series thus gives us a decent approximation to the final answer.

Ignoring the details of the math, it was easy to get a good approximation for this particular product because the terms in your product converge geometrically to 1. If they converge slowly, then you will have to multiply far more terms to get a decent approximation.
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