Sum of 1/n^2+1
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a worked example of summing series like 1/(n^2+1) with Fourier series and Parseval's identity, including variants of the Basel problem.
Recommended by 1 people across 1 different months, 2019 to 2022.
What it is
Dr Peyam works out the sum of 1/(n^2+1) from n=0 to infinity in closed form. He finds the complex Fourier coefficients of e^x and applies Parseval's identity to get the result, checking whether arctan or pi^2/6 appears.
What people said
Recommends PrincessEev · 2019-01 · Mathematics Stack Exchange · on “General formula for $\sum_{n=0}^\infty \frac{1}{(an+b)^c}, a> 0, b>0, c>1$”
With respect to one particular case ($a=b=1, c=2$), there is a discussion on it in this video - [this video] - I haven't watched it yet so I don't have anything to offer there. Just figured you might find it useful.
Cites guavas222 · 2021-06 · Mathematics Stack Exchange · on “The Double Basel Problem”
I have been playing with the series which I had been calling the 'Double Basel problem' for the past couple of hours $$ \sum_{n=1}^{\infty} \sum_{m=1}^\infty \frac{1}{{n^2 +m^2}}. $$ After wrestling with this for awhile, I managed to generalize a result demonstrated here [this video] . This identity is: $$ \sum_{m=1}^{\infty}\frac{1}{x^2+m^2} = \frac{1}{2x}\left[ \pi \coth{\pi x} - \frac{1}{x}\right]. $$ Hence the original series becomes: $$ \sum_{n=1}^{\infty} \frac{1}{2n}\left[\pi \coth{\pi n} - \frac{1}{n} \right]. $$ I have no idea where to go next with this problem. I seriously doubt that this series is convergent; however, I have been unable to prove it. Can you prove that this series is di
Cites user10478 · 2022-02 · Mathematics Stack Exchange · on “Parseval's Identity Application”
In this video [this video] , it is stated that Parseval's Identity "is how we go from discrete to continuous." However, I have not been able to find any material that expands on this use of Parseval's Identity. I have discovered that it is the inner product version of the Pythagorean Theorem. What does this quote that Parseval's Identity "is how we go from discrete to continuous" mean? Is Parseval's Identity a bridge between recurrence relations and differential equations somehow? In this vein, is it related to time-scale calculus [LINK] or this application of generating functions [LINK] ?