Answer to Question #130705 in Complex Analysis for Penny

Question

Infinite product (1+Zn) converges
Show that lim Zn=0

Answer 1

Based on the definition of convergent series, the following theorem is proved in complex analysis:

If $∏^∞_{n=0} z_n$ converges, then $\lim_{n→∞}{z_n}= 1$.

Let's call $\tilde{z}_n = (1 +z_n).$ By the condition of the task:

$∏^∞_{n=0} \tilde{z_n} = ∏^∞_{n=0} (1+z_n)$ converges $\Leftrightarrow \lim_{n→∞}{\tilde{z}_n}= 1$

$\lim_{n→∞}{(1+z_n)}= 1$

$\lim_{n→∞}{1}+\lim_{n→∞}{z_n}= 1$

$1+\lim_{n→∞}{z_n}= 1$

$\lim_{n→∞}{z_n}= 0$ Q.E.D.