I'm trying to find a function corresponding to the following Taylor series: $\sum_{n=2}^\infty\frac{x^n}{n(n-1)}$. I found it to converge if $-1\leq x \leq 1$, but I'm not sure how to find the function corresponding to this. What are some good techniques for problems like this?
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$\begingroup$As you saw in the comments already, you can differentiate the series, $f(x) = \sum_{n=2}^\infty\frac{x^n}{n(n-1)}$ twice to see $f''(x) = \sum_{n=2}^\infty x^{n-2}$. You can now do a substitution of $m=n-2$ to see $f''(x) = \sum_{m=0}^\infty x^m$. This is the geometric series $g(x) = \frac{1}{1-x}$. From this, you can integrate twice to find $f(x)= x + \ln(1-x)-x\ln(1-x)$.
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