an infinite product

We are all comfortable with adding infinitely many numbers — that is a series. An infinite product is the multiplicative cousin: instead of a running total you build a running product. Write the factors as 1 + a_n, so the product is (1 + a_1)(1 + a_2)(1 + a_3) ... and you watch the partial products P_N = (1 + a_1) ... (1 + a_N) as N grows. The natural question is the same one we ask of series: do these partial products settle down to a single finite number?

There is one twist that has no analogue for sums, and it is the whole reason products are written as 1 + a_n rather than just b_n. With addition, the only way to 'settle' is to add ever-smaller pieces; with multiplication you can also settle by multiplying by numbers ever-closer to 1. So the right notion of a convergent infinite product is that P_N tends to a NONZERO limit P (we usually exclude the limit 0 from 'convergence', otherwise a single zero factor would let any garbage afterward look convergent). For the partial products to approach 1's behaviour the factors must approach 1, i.e. a_n must tend to 0 — that is the multiplicative version of the n-th term test.

Why bother building functions by multiplying rather than adding? Because zeros multiply. A polynomial is c(z - r_1)(z - r_2) ... (z - r_n) — a finite product whose factors vanish exactly at its roots. Infinite products are how we carry that beautiful factored picture to functions with infinitely many zeros, like sin(pi z), which we will write as an infinite product whose factors vanish at every integer. Products put the zeros of a function on display the way a sum never can.