Asymptotic expansion in 0 of a series

According to Find the sum of $\sum \frac{1}{k^2 - a^2}$ when $0<a<1$, the following identity holds $$\sum_{n=1}^\infty \frac1{n^2-z^2}=\frac1{2z^2}-\frac{\pi\cot\,\pi z}{2z}.$$ After letting $z=ix$, we find $$\sum_{n=1}^\infty \frac1{n^2+x^2}=\frac{\pi\coth(\pi x)}{2x}-\frac1{2x^2}.$$ Finally, after expanding the Hyperbolic Cotangent at $x=0$, we get $$\sum_{n=1}^\infty \frac1{n^2+x^2}=\frac{\pi^2}{6}-\frac{\pi^4}{90}x^2 +O(x^4).$$ Notice that $\frac{\pi^4}{90}=\sum_{n=1}^{\infty}\frac{1}{n^4}$: $$\sum_{n=1}^\infty \frac1{n^2+x^2}=\sum_{n=1}^\infty \frac{1}{n^2}\frac1{1+(x/n)^2}=\sum_{n=1}^\infty \frac{1}{n^2}\left(1-\frac{x^2}{n^2}+O(x^4)\right)\\=\sum_{n=1}^\infty\frac1{n^2}-\sum_{n=1}^\infty\frac1{n^4}\,x^2+O(x^4).$$