Evaluate $\sum\limits_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}$

How to find $$\sum_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}$$

I try something like this:

$$\begin{align*}\sum_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}=\sum_{k=1}^{\infty}\frac{k^2}{k^4+k^2+1}-\sum_{k=1}^{\infty}\frac{1}{k^4+k^2+1}.\end{align*}$$

Using fact that $$\sum_{k = 1}^{n}{\frac{1}{k^4+k^2+1}}=\frac{1}{2}\cdot\frac{n+1}{n^2+n+1}+\frac{1}{2}\cdot\sum_{k = 1}^{n-1}{\frac{1}{k^2+k+1}}$$ we find that $$\begin{align*}\sum_{k=1}^{\infty}\frac{1}{k^4+k^2+1} &=\frac{1}{2}\cdot\sum_{k=1}^{\infty}{\frac{1}{k^2+k+1}}\\ &=\frac{1}{6}\left(\sqrt{3}\pi \tanh{\left(\frac{\sqrt{3}\pi}{2}\right)}-1\right).\end{align*}$$

But I don't know how to find $\displaystyle\sum_{k=1}^{\infty}\frac{k^2}{k^4+k^2+1}.$

If someone want to know how to evaluate $\displaystyle\sum_{k=0}^{\infty}\frac{1}{k^2+k+1}$:

First, $$\displaystyle\sum_{k=0}^{\infty} \frac{1}{k^2+k+1}=\sum_{k=0}^{\infty}{\frac{1}{\left(k+\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}}.$$ Now, using "well-know" formula $$\displaystyle\cos(\phi)=\prod_{k=0}^{\infty}{\left( 1-\frac{4\phi^2}{(2k+1)^2\pi^2}\right)}$$ we find that $$\displaystyle\log (\cos(\phi))=\sum_{k=0}^{\infty}{\log\left( 1-\frac{4\phi^2}{(2k+1)^2\pi^2}\right)}$$ and then we attack with $\dfrac{d}{d\phi}$ and find $$\displaystyle\tan(\phi)=\sum_{k=0}^{\infty}{\frac{8\phi}{(2k+1)^2\pi^2-4\phi^2}}.$$ Let $\phi=\pi\alpha\cdot i$, then we get $$\displaystyle\tan(\pi\alpha\cdot i)=i\cdot\tanh(\pi\alpha)=i\cdot\sum_{k=0}^{\infty}{\frac{8\pi\alpha}{(2k+1)^2\pi^2+4\pi^2\alpha^2}}=\frac{2\alpha i}{\pi}\cdot\sum_{k=0}^{\infty}{\frac{1}{\left(k+\frac{1}{2}\right)^2+\alpha^2}}.$$ So, we find that $$\displaystyle\sum_{k=0}^{\infty}{\frac{1}{\left(k+\frac{1}{2}\right)^2+\alpha^2}}=\frac{\pi}{2\alpha}\cdot\tanh(\pi\alpha).$$ Let $ \alpha=\dfrac{\sqrt{3}}{2}.$ We get $$\displaystyle\sum_{k=0}^{\infty}{\frac{1}{\left(k+\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}}=\frac{\sqrt{3}\pi}{3}\cdot\tanh\left(\frac{\sqrt{3}\pi}{2}\right)$$ or $$\displaystyle\sum_{k=0}^{\infty} \frac{1}{k^2+k+1}=\frac{\sqrt{3}\pi}{3}\cdot\tanh\left(\frac{\sqrt{3}\pi}{2}\right).$$

Solution 1:

Using Partial Fraction Decomposition $$\frac{k^2-1}{k^4+k^2+1}=\frac{Ak+B}{k^2-k+1}+\frac{Ck+D}{k^2+k+1}$$

So, $k^2-1=k^3(A+C)+k^2(A+B-C+D)+k(A+B+C-D)+B+D$

Comparing the coefficients of different powers of $k$ in the above identity,

$A+C=0$

From $A+B+C-D=0,B+D=0$ and $B+D=-1\implies B=-D=-\frac12$

From $A+B-C+D=1\implies A-C=2$ and $A+C=0\implies A=-C=1$

$$\implies\frac{k^2-1}{k^4+k^2+1}=\frac{k-\frac12}{k^2-k+1}-\frac{k+\frac12}{k^2+k+1}$$

$$\frac{2(k^2-1)}{k^4+k^2+1}=\frac{2k-1}{k^2-k+1}-\frac{2k+1}{k^2+k+1}=T(k)\text{ say}$$

$$\implies T(n)=\frac{2n-1}{n^2-n+1}-\frac{2n+1}{n^2+n+1}$$

If we set $U(m)=\dfrac{2m-1}{m^2-m+1},U(m+1)=\dfrac{2(m+1)-1}{(m+1)^2-(m+1)+1}=\dfrac{2m+1}{m^2+m+1}$

$$\implies T(n)=U(n)-U(n+1)$$

Clearly, the first part of any term except the first term is cancelled by the last part of the previous term.

$$\implies2\sum_{k=1}^{\infty} \frac{k^2-1}{k^4+k^2+1}=U(1)=\cdots$$