Intereting Posts

Two problems on number theory
Finding Markov chain transition matrix using mathematical induction
Intuition surrounding units in $R$
Fractional Calculus: Motivation and Foundations.
Parabola and a line
About the asymptotic formula of Bessel function
A nicer proof of Lagrange's 'best approximations' law?
Is a Sudoku a Cayley table for a group?
Evaluating $\sum_{k=1}^{\infty}\frac{\sin\left(k\theta\right)}{k^{2u+1}}$ with multiple integrals
Probability of first and second drawn balls of the same color, without replacement
The continuity assumption in Schwarz's reflection principle
Find $\lim_{x\to 1}\frac{p}{1-x^p}-\frac{q}{1-x^q}$
Could we show $1-(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots)^2=(1-\frac{x^2}{2!}+\frac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?
Probability of a random binary string containing a long run of 1s?
Find the LCM of 3 numbers given HCF of each 2.

How do I find:

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

I am pretty sure that the infinite product converges, but if it doesn’t please let me know if I have made an error.

Also, could I have a nice explanation as to how would someone arrive to the answer.

- Compact but not sequentially compact question
- Proving completeness of a metric space
- Ratio test and the Root test
- Continuity of a series of functions
- Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric
- Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Thanks alot.

- Measuring the set-theoretical complexity of sets/spaces encountered in general analysis
- Continuation of smooth functions on the bounded domain
- Show that the linear operator $(Tf)(x)=\frac{1}{\pi} \int_0^{\infty} \frac{f(y)}{(x+y)} dy$ satisfies $\|T\|\leq 1$.
- Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$
- Prove that the function is continuous and differentiable (not as easy as it sounds imo)
- Proving $d$ is a metric of a power set
- Composition of Riemann integrable functions
- How can I find $\sum\limits_{n=0}^{\infty}\left(\frac{(-1)^n}{2n+1}\sum\limits_{k=0}^{2n}\frac{1}{2n+4k+3}\right)$?
- How to prove that $\,\,f\equiv 0,$ without using Weierstrass theorem?
- Limit of a function. Integration problem

An infinite product $\prod_{n \ge 0} (1 + u_n)$ where $u_n > 0$ converges if and only if $\sum_{n \ge 0} u_n$ converges, and $\sum_{n \ge 1} \frac{1}{n^2} = \frac{\pi^2}{6}$ is a famous result by Euler.

Euler’s product formula for $\sin z$ is hard to prove, but intuitive (it has roots at $\pm n \pi$):

$$

\frac{\sin z}{z} = \prod_{n \ge 1} \left( 1 – \frac{z^2}{\pi^2 n^2} \right)

$$

This gives directly $\dfrac{\sin i}{i} = \dfrac{e – e^{-1}}{2}$ for your product

The product converges because $\sum n^{-2}$ does. Can you see why? What is the greatest sum that can possibly appear when we expand the product?

In fact, suppose that $a_n\geq 0$ for each $n$. Set $$p_n=\prod_{k=1}^n a_k$$

Then $\log p_n=\sum_{k=1}^n\log a_k$

If $\sum a_k$ converges, then $a_k\to 0$, then since $$\lim_{x\to 0}\frac{\log (1+x)}x=1$$ so that $$\lim \frac{\log (1+a_n)}{a_n}=1$$

the comparison test tells us $\log p_n$ converges, say to $\ell$. By continuity of the logarithm, $p_n$ must converge to $p$ with $\log p=\ell$ so that $\lim p_n=e^\ell$.

Conversely, suppose $\log p_n=\sum_{k=1}^n\log a_k$ converges. This means that $\log(1+a_k)\to 0$ so that $a_k\to 0$. Comparison yields from $$\lim \frac{\log (1+a_n)}{a_n}=1$$ that $$\sum a_k$$ converges too. Thus, we have shown

**PROP** If $a_n\geq 0$ then $\prod (1+a_k)$ converges if and only if $\sum a_k$ (if and only if $\sum \log(1+a_k)$ does.)

- A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$
- Show that the conditional statement is a tautology without using a truth table
- What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$
- Finitely generated idempotent ideals are principal: proof without using Nakayama's lemma
- Unit square as union of two simplexes
- What happens if I toss a coin with decreasing probability to get a head?
- Giving an asymptotically tight bound on sum $\sum_{k=1}^n (\log_2 k)^2$
- Solving an ordinary differential equation with initial conditions
- Numerical method for finding the square-root.
- How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$
- Trace of the power matrix is null
- Why does this Fourier series have a finite number of terms?
- Deductive proof – need help, explanation how to
- How are the cardinalities of the object images of adjoint functors related?
- Angle between 2 points