Prove $\prod_1^\infty (1+p_n)$ converges

Question

Let $p_{2n-1} = \frac{-1}{\sqrt{n}}$, and $p_{2n} = \frac{1}{n}+\frac{1}{\sqrt{n}}$.

Prove $\prod_1^\infty (1+p_n)$ converges.

By numerical simulations, it appears to converge (to something around $0.759$). However, I'm not sure how to prove this. I know we can skip the first term since it's $0$. Then we can write it in the following form.

\begin{align*} \prod_1^\infty (1+p_n) &= \prod_1^\infty \left(1+\frac{1}{2n}+\frac{1}{\sqrt{2n}}\right)\left(1-\frac{1}{\sqrt{2n+1}}\right) \\ &= \left(1+\frac{1}{2}+\frac{1}{\sqrt{2}}\right)\left(1-\frac{1}{\sqrt{3}}\right)\left(1+\frac{1}{4}+\frac{1}{\sqrt{4}}\right)\left(1-\frac{1}{\sqrt{5}}\right)... \end{align*} Any thoughts?

Answer 1

Note that

$$\prod\limits_{k=2}^{2n}(1+p_k) = \prod\limits_{k=2}^{n}(1+p_{2k-1})(1+p_{2k}) = \prod\limits_{k=2}^{n}\left(1-\dfrac{1}{\sqrt{k}}\right)\left(1+\dfrac{1}{k} +\dfrac{1}{\sqrt{k}}\right) = \prod\limits_{k=2}^{n} \left(1- \dfrac{1}{k\sqrt{k}}\right)$$

It is also known that for any sequence $\{a_k\}$ such that $\forall k \geqslant 2: 0 \leqslant a_k < 1$:

$$\prod\limits_{k=2}^{n} \left(1- a_k\right) \leqslant e^{-\sum\limits_{k=2}^n a_k}$$

The inequality above is true because $1 -x \leqslant e^{x}$ for $0 \leqslant x < 1$.

Hence, for $a_k = \dfrac{1}{k\sqrt{k}}$:

$$\prod\limits_{k=2}^{n} \left(1- \dfrac{1}{k\sqrt{k}}\right) \leqslant \exp\left(-\sum\limits_{k=2}^n \dfrac{1}{k\sqrt{k}}\right)$$

Thus, to finish the proof it is enough to note that $\sum\limits_{k=2}^n \dfrac{1}{k\sqrt{k}}$ converges.

Answer 2

I'm not 100% sure if this can be made rigorous, but maybe you can take it as a starting point.

\begin{align*}\log \Pi(1+p_n)= \sum \log((1+\frac{1}{2n}+\frac{1}{\sqrt{2n}})(1-\frac{1}{\sqrt{2n+1}}))\\\approx\sum \log((1+\frac{1}{\sqrt{2n}})(1-\frac{1}{\sqrt{2n}}))\approx\sum \log(1-\frac{1}{2n})\approx \sum -\frac{1}{2n} \rightarrow \infty\end{align*}

using $\log(1+\frac{1}{n})\approx \frac{1}{n}$

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Actually there appears to be a cancellation that makes it so I can't neglect $1/2n$ the way I did. $(1+\frac{1}{2n}+\frac{1}{\sqrt{2n}})(1-\frac{1}{\sqrt{2n+1}}) =1+(\frac{1}{\sqrt{2n}}-\frac{1}{\sqrt{2n+1}})+(\frac{1}{2n}-\frac{1}{\sqrt{(2n)(2n+1)}})-\frac{1}{2n\sqrt{2n+1}}$ and asymptotically the first grouped terms are $\approx \frac{1}{2(2n)^{3/2}}$ ... in short, the product converges.