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I am reading this article about the Riemann Hypothesis and it states:

Lemma 1.Suppose ${ (a_n) }$ is a series. If $\sum_{n=1}^\infty a_n < \infty$ , then the product $\prod_{n=1}^\infty (1+ a_n)$ converges. Further, the product converges to 0 if and only if one if its factors is $0$.

I wonder how can one proof it? Also I am surprised about the fact that when one $a_n$ is $0$ that then the product converges to $0$. If I imagine that every $a_n$ is $0$, then I would assume that the product converges to $1$ ?

Adam
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1 Answers1

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Take the logarithm of the product:

$$\log\left(\prod_{n=1}^\infty(1+a_n)\right)=\sum_{n=1}^\infty\log(1+a_n)$$

for all $\;n\;$ big enough, $\;|a_n|<\frac14\;$ , say, so the sereis above is a positive one and

$$\frac{\log(1+a_n)}{a_n}\xrightarrow[n\to\infty]{} 1$$

and the limit comparison test gives you the convergence

Can you finish now the argument?

DonAntonio
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  • Okay maybe I was not able to finish the argument. I see that we have $log(1+a_n) < C \cdot |a_n| $ for large enough $n$. If $a_n$ was absolutely convergent, then we would know the same about $ log(1+a_n)$. How do you finish it if $ a_n$ is not absolutely convergent? – Adam May 07 '18 at 22:08