<s>[INST] How can I find the sum of $\sum_{n=2}^\infty \frac{(-1)^n}{n^2 - n}$ How can I find the sum of the series:
$$\sum_{n=2}^\infty \frac{(-1)^n}{n^2 - n}$$
I did this:
$$\sum_{n=2}^\infty \left(\frac{(-1)^n}{n-1} + \frac{(-1)^{n-1}}{n}\right)$$
But I am not sure how to proceed.
 [/INST] Let us consider a sequence
$$a_n = \sum_{k=n+1}^{2n} \frac{1}{k}$$
Now, let's attempt to get a recursive formula for this.
$$a_n - a_{n-1} = \sum_{k=n+1}^{2n} \frac{1}{k} - \sum_{k=n}^{2n-2} \frac{1}{k} = -\frac{1}{n} + \frac{1}{2n-1} + \frac{1}{2n} = \frac{1}{(2n-1)(2n)}$$
$$a_n = a_{n-1} + \frac{1}{(2n-1)(2n)}$$
Since $a_1 = \frac{1}{2} = \frac{1}{(2\cdot1 - 1)(2\cdot1)}$,
$$a_n = \sum_{k=1}^n \frac{1}{(2k-1)(2k)}$$
Now if we also consider the sequence alternating harmonic series
$$S_n = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{(-1)^{n-1}}{n}$$
We can see that
$$S_{2n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{1}{2n - 1} - \frac{1}{2n}$$
$$S_{2n} = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots + \left(\frac{1}{2n - 1} - \frac{1}{2n}\right)$$
$$S_{2n} = \sum_{k=1}^n \left(\frac{1}{2k-1} - \frac{1}{2k}\right) = \sum_{k=1}^n \frac{1}{(2k-1)(2k)}$$
This means that $S_{2n} = a_n$. If we now look at the expansion of $a_n$
$$a_n = \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + \cdots + \frac{1}{2n}$$
If we factor out $\frac{1}{n}$, we get
$$a_n = \frac{1}{n} \left(\frac{1}{1+\frac{1}{n}} + \frac{1}{1+\frac{2}{n}} + \frac{1}{1+\frac{3}{n}} + \cdots + \frac{1}{1+\frac{n}{n}}\right)$$
As $n$ tends towards infinity, this is the Riemann sum for $\int_0^1 \frac{1}{1+x}~{\rm d}x$.
$$\int_0^1 \frac{1}{1+x}~{\rm d}x = \int_1^2 \frac{1}{x}~{\rm d}x = \ln 2$$
You can then use that to solve your sum.

Alternative: Consider successive derivatives of $\ln(1 + x)$.
$$\frac{{\rm d}}{{\rm d}x} \left[\ln(1 + x)\right] = \frac{1}{1 + x} = \left(1 + x\right)^{-1}$$
$$\frac{{\rm d}^2}{{\rm d}x^2} \left[\ln(1 + x)\right] = -\left(1 + x\right)^{-2}$$
$$\frac{{\rm d}^3}{{\rm d}x^3} \left[\ln(1 + x)\right] = 2\left(1 + x\right)^{-3}$$
$$\frac{{\rm d}^4}{{\rm d}x^4} \left[\ln(1 + x)\right] = -6\left(1 + x\right)^{-4}$$
$$\vdots$$
$$\frac{{\rm d}^n}{{\rm d}x^n} \left[\ln(1 + x)\right] = (-1)^{n-1} \cdot (n-1)! \cdot \left(1 + x\right)^{-n}$$
Using this to create a Taylor series about $x=0$ gives us
$$\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots + \frac{(-1)^{n-1} \cdot x^n}{n} + \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} \cdot x^n}{n}$$
Plugging in $x=1$ to give the alternating harmonic series gives us
$$\ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + \frac{(-1)^{n-1}}{n} + \cdots = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}$$

Now knowing that the alternating harmonic series is equal to $\ln 2$, we can solve the sum.
$$\sum_{n=2}^\infty \left(\frac{(-1)^n}{n-1} + \frac{(-1)^{n-1}}{n}\right)$$
$$= \sum_{n=2}^\infty \frac{(-1)^n}{n-1} + \sum_{n=2}^\infty \frac{(-1)^{n-1}}{n}$$
$$= \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} + \left(-1 + \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}\right)$$
$$= \ln (2) + \ln (2) - 1 = \ln (4) - 1$$
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