Return to Question

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result : $$\sum_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{1+ \pi\coth\pi}{2}$$ is known by WolframAlpha. But my question is : does there exists a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result : $$\sum_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{1+ \pi\coth\pi}{2}$$ is known by WolframAlpha. But my question is : does there exist a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result : $$\sum_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{\pi+ \coth\pi}{2}$$ is known by WolframAlpha. But my question is : does there exists a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result : $$\sum_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{1+ \pi\coth\pi}{2}$$ is known by WolframAlpha. But my question is : does there exists a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?

Actually, as the corresponding integral ($\ln(\cos x)/(1-x)$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result : $\sum_{n=0}^{+\infty}\frac1{n^2+1}=(\pi+ \coth\pi)/2$ is known by WolframAlpha. But my question is : does there exists a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac1{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it shown somewhere using Fourier series. By the way, the result : $$\sum_{n=0}^{\infty}\frac{1}{n^2+1}= \frac{\pi+ \coth\pi}{2}$$ is known by WolframAlpha. But my question is : does there exists a general theory of similar results (something like exact formulas for $\sum_{n=0}^{+\infty}\frac{1}{an^2+bn+c}$, at least when $a$, $b$ and $c$ are integers)?