I am reading about a convergence problem with infinite products and I am told:

Any finite sequence $\{c_{n}\}$ in the complex plane has an associated polynomial p(z) that has zeroes precisely at the points of that sequence, $p(z)=\,\prod _{n}(z-c_{n})$.

Also, any polynomial function p(z) in the complex plane has a factorization $\,p(z)=a\prod _{n}(z-c_{n})$, where a is a non-zero constant and $c_{n}$ are the zeroes of p.

It then says that the infinite product does not converge:

When one considers the product $\,\prod _{n}(z-c_{n})$ when the sequence $\{c_{n}\}$ is not finite then it can never define an entire function, because the infinite product does not converge.

It is somewhat unclear to me why it does not converge, but in any case, it then goes on to state that:

A necessary condition for convergence of the infinite product in question is that each factor $(z-c_{n})$ must approach 1 as $n\to \infty$

This last part flew right over my head. Why is it necessary for $(z-c_{n})$ to approach 1? There's no explanation provided. I must be lacking some fundamental knowledge so can someone explain to me in Layman's Terms why this rule holds true, or perhaps show a proof that might help me understand?

  • $\begingroup$ This is similar to the term test for an infinite series. If the series $\sum_{n \geqslant 1}a_n$ converges then it is necessary for $\lim_{n \to \infty} a_n = 0$ since the general term is the difference between two partial sums converging to the same value. An infinite series is a sequence of partial sums, an infinite product is a sequence of partial products. So if the infinite product converges then the general term, as a ratio of partial products converging to the same value, must converge to $1$. $\endgroup$ – RRL Mar 24 at 8:44

If the infinite product is convergent, i.e., $P_n = \prod_{k=1}^n(z - c_k) \to P$ as $n \to \infty$, then

$$\lim_{n \to \infty} (z- c_n) = \lim_{n \to \infty}\frac{P_n}{P_{n-1}} = \frac{\lim_{n \to \infty}P_n}{\lim_{n \to \infty}P_{n-1}} =1$$


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