Infinite multiplication

[PrayingMantis]

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More generally, it will happen if the sequence of numbers that you are multiplying doesn't converge to 1, or converges slowly to 1. If the numbers in your sequence go to 1 quickly enough, then M can be non-zero.

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Thanks for the answer. Can you please elaborate a bit on how is the "quickness" of the convergence effecting M? that is, what is the criteria for a sequence that is converging to 1, which for it M=0, as opposed to a different sequence that also converges to 1, but that produces a non-zero M?

There is no simple criterion that is necessary and sufficient, but I can give some examples. If the n-th term in your sequence is e^{-1/n}, then the log is -1/n, so the sum of the logs diverges (it is the negative harmonic series) and M=0. If the n-th term in your sequence is e^{-1/n^2}, then the log is -1/n^2, and so the sum of the logs converges and M is not 0. IIRC, the sum of -1/n^2 is -pi^2/6 [aka -zeta(2)], and so M=e^[-pi^2/6] for that example.

There are a variety of tests for whether or not a series converges--see your favorite calculus textbook. One trick that might be useful in dealing with examples is that log(1-x) ~ -x for small x.