the convergence of an infinite product
How do you tell whether an infinite product (1 + a_1)(1 + a_2) ... actually settles down? The honest answer turns multiplication into addition by taking logarithms. If you take the log of a product you get a sum: log[(1+a_1) ... (1+a_N)] = log(1+a_1) + ... + log(1+a_N). So the product converges (to a nonzero limit) precisely when the series sum log(1 + a_n) converges. Convergence of products is convergence of series in disguise — you trade a hard multiplicative question for the familiar additive one.
Because log(1 + a) is close to a when a is small (the Taylor expansion log(1 + a) = a - a^2/2 + ... ), the clean working criterion is this: the product of (1 + a_n) converges ABSOLUTELY exactly when the series sum |a_n| converges. This is the test you actually use. It says: ignore the logarithms, just check whether the sum of the moduli of the little deviations a_n is finite. If sum |a_n| < infinity, the product converges to a nonzero value, and — crucially — you may reorder the factors freely without changing the answer, just as with an absolutely convergent series.
Two cautions keep you honest. First, you need the factors eventually nonzero for the logarithm step to make sense; finitely many genuinely-zero factors are allowed and simply make the product 0 (which is how zeros of a function show up). Second, this criterion is about absolute convergence; one can build products that converge only conditionally, where the order of the factors matters, but in function theory we almost always engineer absolute convergence so we can manipulate freely. The slogan to remember: sum |a_n| < infinity is the safe harbour for infinite products.
Consider the product of (1 + z^n / 2^n) for n >= 1 at a fixed z with |z| < 2. Here a_n = z^n / 2^n, so |a_n| = (|z|/2)^n is a geometric series with ratio < 1, hence sum |a_n| < infinity. The product therefore converges absolutely. By contrast the product of (1 + 1/n) diverges, because sum 1/n = infinity — its partial products grow like N + 1 without bound.
Absolute convergence of a product reduces to summability of |a_n|; a geometric tail converges, the harmonic one does not.
A common trap: the product of (1 - 1/n) for n >= 2 has all factors strictly between 0 and 1 and each close to 1, yet it tends to 0, not a positive limit — because sum 1/n diverges. Tending to 0 is counted as DIVERGENCE for products, exactly to keep the theory clean.