Prove $\displaystyle\prod_{n=1}^\infty (1 + a_n)$ converges if and only if $\displaystyle\sum_{n=1}^\infty a_n$ converges.
Note: There are proofs of this readily available. My question is to point me in the direction for myself to prove it. Please do not finish my proof. Instead, please tell me if I'm going in the right direction or, if not, provide gentle assistance.
I'm having trouble completing this proof, and would like help taking the next step. Please do not give away the full proof. Below are my approaches.
Given $(a_n)$, let $s_n = \sum_{n=1}^\infty a_n$ and $p_n = \prod_{n=1}^\infty (1 + a_n)$. To simplify, I will first suppose $a_n \geq 0$ for all $n$.
Approach 1:
Suppose $(s_n)$ converges. For any $a > 0$, there exists a $k$ such that $a_n < a$. Then, $p_n \leq p_k(1 + a)^{n-k}$ for $n \geq k$. Can we show $\lim_{n \to \infty}(1 + a)^{n-k}$ converges?
I tried applying the binomial theorem, or taking the log, neither of which simplified this.
I imagined taking a continuous extension, giving the indeterminate form $(1 + 0)^\infty$, so I could apply L'Hopital's rule, but made no progress.
Approach 2: Replace the infinite product with an infinite sum of $\log (1 + a_n)$ and show this converges. No progress here either.
Approach 3: $p_{n+1} = p_n + p_n a_n$. I tried to use this to form a sequence or series that I could show converges.
Approach 4: $p_{n+1} = p_n(1 + a_n)$. If I knew $(p_n)$ was bounded, I believe I could use this to show convergence. But I could not prove $(p_n)$ was bounded.
Best Answer
Hint: Look at $$ \lim\limits_{x\rightarrow 0} \frac{\log(1+x)}{x} $$
Your four approaches are all very reasonable. It just takes some tinkering to find out which one will ultimately get you the right answer. I think approach 3 could be used to prove that convergence of the product implies convergence of the sum, but the other direction would be tough. However, you should be able to get both directions from approach 2.