What is the answer of $\prod_{n=2}^{\infty} (1+\dfrac{1}{n^2}) = \frac 5 4 \times \frac{10}{9} \times \frac{17}{16} \times \cdots$

Question

What is the answer of

$$\prod_{n=2}^{\infty} \left( 1+\frac{1}{n^2}\right)$$

which is

$$\frac 5 4 \times \frac{10}{9} \times \frac{17}{16} \times \cdots$$

This question is from a test-paper of my little sister who is in high school and the hint provided are:

$$\ln(x) < x-1$$

$$\sum_{n=1}^\infty \frac1{n^2}= \frac{\pi^2}{6}$$

My current work:

$$\prod_{n=2}^{\infty} \left( 1+\frac{1}{n^2}\right) < \prod_{n=2}^{\infty} \left( 1+\frac{1}{n^2-1}\right) = \prod_{n=2}^{\infty} \frac{n\cdot n}{(n+1)(n-1)} = 2$$

$$ \prod_{n=2}^{\infty} \left( 1+\frac{1}{n^2}\right) < \prod_{n=2}^{\infty} \mathrm{e}^{1/n^2} = \mathrm{e}^{\pi^2/6-1} \approx 1.906 $$

@Gary is right, it is $\dfrac{e^\pi - e^{-\pi}}{4\pi}$, I am thinking of a high-school solution. — Firestar-Reimu, Apr 04 '22 at 03:19
Strikes me as unlikely that high school math would lead to an answer like that, unless it's a very advanced high school. — Gerry Myerson, Apr 04 '22 at 03:32
Is https://math.stackexchange.com/questions/3948170/finding-the-value-of-this-infinite-product elementary enough? — Gerry Myerson, Apr 04 '22 at 03:37
Here's a derivation that uses Stirling's formula, applied to complex numbers: https://math.stackexchange.com/questions/2994156/prove-that-prod-k-2-infty-11-k2-sinh-pi-2-pi — Gerry Myerson, Apr 04 '22 at 03:44
@GerryMyerson: I mad a six years too late answer to https://math.stackexchange.com/questions/2039393/what-is-the-value-of-prod-n-1-infty-1-frac1n2/4419751#4419751 for the partial product. — Claude Leibovici, Apr 04 '22 at 04:45

Firestar-Reimu · Answer 1 · 2022-04-05T01:06:17.227

1

Thanks @Gary

from:

$$ \sinh(z) = z\prod_{n=1}^{\infty} \left(1+\frac{z^2}{n^2 \pi^2}\right) $$

with $z=\pi$, we get:

$$ \prod_{n=2}^{\infty} \left( 1+\frac{1}{n^2}\right) = \frac{\sinh \pi}{2\pi} \approx 1.82804 < 2 $$

edited Apr 05 '22 at 01:06

answered Apr 04 '22 at 03:32

Firestar-Reimu

191

Is that a high school solution? Can you get that infinite product for the hyperbolic sine using high school methods? – Gerry Myerson Apr 04 '22 at 03:38
I will edit this question. This solution is from @Gary and this is not a high-school method. – Firestar-Reimu Apr 04 '22 at 15:59
Where does the $\frac1{2\pi}$ come from? – Thomas Andrews Apr 04 '22 at 16:17
You forgot the $z$ term in the product. $\sinh(z)=z\prod_1^\infty\dots$ But then the denominator should be $\pi,$ not $2\pi?$ – Thomas Andrews Apr 04 '22 at 16:19
@ThomasAndrews it is a typo, sorry. – Firestar-Reimu Apr 05 '22 at 01:06

What is the answer of $\prod_{n=2}^{\infty} (1+\dfrac{1}{n^2}) = \frac 5 4 \times \frac{10}{9} \times \frac{17}{16} \times \cdots$

1 Answers1