Can someone help me with this equality?
Prove that $$\sum_{i=1}^n \frac{x^i}{i} = \sum_{i=1}^n {n \choose i}\frac{(x-1)^i}{i} + H_n$$ where $H_n$ is the harmonic number.
2026-04-01 16:27:48.1775060868
Two ways to calculate the same sum where the harmonic number pops up.
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Hint: let the LHS be $L(x)$ and the RHS be $R(x)$. For proving $L(x)=R(x)$, it's sufficient to check that $L'(x)=R'(x)$ (done using the binomial theorem) and $L(1)=R(1)$ (done immediately).