<s>[INST] Help for series calculation $\sum_{n\ge1} \frac{1}{4n^3-n}$ I want to find the following series.
$$\sum_{n\ge1} \frac{1}{4n^3-n}$$ 
So, fisrt of all, patial fraction : $$\frac{1}{4n^3-n}=\frac{1}{2n+1}+\frac{1}{2n-1}-\frac{1}{n}.$$
Next, I consider the geometric series $$\sum_{n\ge1} x^{2n-2}+x^{2n}-x^{n-1}=\frac{1}{1-x^2}+\frac{x^2}{1-x^2}-\frac{1}{1-x}\tag{1}$$ where ${-1\lt x\lt1}$.
Then, I integrate both side of (1) from $0$ to $1$:
$$\sum_{n\ge1}\frac{1}{2n-1}+\frac{1}{2n+1}-\frac{1}{n}=\int_0^1 \frac{x^2-x}{1-x^2}dx$$
Finally, I get the value that is equal to $ln(2)-1$  by improper integral but it's less than $0$. What did I do wrong? All help is appreciated.
 [/INST] Following a comment of mine above:


*

*without checking too carefully, the issue in your step most likely resides in the "I integrate both sides of (1) from 0 to 1." You then integrate the LHS termwise -- why can you do that? The interval of convergence of your power series is $1$, so it is not obvious you can swap $\sum_n$ and $\int_0^1$. And, as it turns out, worse than "not obvious" it appears to be wrong.

*Now, another approach to compute the sum: starting with your $(1)$, we write, for $N\geq 1$,
$$\begin{align}
\sum_{n=1}^N a_n &= \sum_{n=1}^N \frac{1}{2n+1}+\sum_{n=1}^N \frac{1}{2n-1}-\sum_{n=1}^N \frac{1}{n}\\
&= \sum_{n=1}^N \left(\frac{1}{2n+1}+\frac{1}{2n}\right)+\sum_{n=1}^N \left(\frac{1}{2n-1}+\frac{1}{2n}\right)-2\sum_{n=1}^N \frac{1}{n} \\
&= \sum_{n=2}^{2N+1} \frac{1}{n}+\sum_{n=1}^{2N} \frac{1}{n}-2\sum_{n=1}^N \frac{1}{n} \\
&= H_{2N+1}-1+H_{2N}-2H_N \\
&= \ln(2N+1)+\ln(2N)-2\ln N - 1 +o(1)\\
&= 2\ln 2 - 1 + \ln(N+\frac{1}{2})+ \ln(N+\frac{1}{2})-2\ln N +o(1)\\
&= 2\ln 2 - 1 + \ln\frac{N+\frac{1}{2}}{N}+ \ln\frac{N-\frac{1}{2}}{N} +o(1)\\
&\xrightarrow[N\to\infty]{} \boxed{2\ln 2 -1}
\end{align}$$
where $H_N = \sum_{n=1}^N \frac{1}{n}$ denotes the $N$-th Harmonic number, which satisfies
$$
H_N = \ln N + \gamma + o(1)
$$
when $N\to\infty$ ($\gamma$ being Euler's constant). We also used that by continuity of the logarithm, $\ln\frac{N\pm\frac{1}{2}}{N} \xrightarrow[N\to\infty]{} \ln 1=0$.
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