Tag Archives: Binomial

Relation between binomial sum and harmonic numbers

Prove that $$\sum_{r=1}^n {n\choose r}(-1)^{r+1}\dfrac{1}{r}=\sum_{r=1}^n \dfrac{1}{r}$$ $$proof$$ Start by $$\sum_{r=0}^n {n\choose r}x^r=(1+x)^n$$ Which can be converted to integration $$\sum_{r=1}^n {n\choose r}\frac{(-1)^{r}}{r}=\int^{-1}_0 \frac{(x+1)^n-1}{x} dx$$ By substitution we have $$\sum_{r=1}^n {n\choose r}\frac{(-1)^{r+1}}{r}=\int^{1}_0 \frac{t^n-1}{t-1} dt = H_n$$ Note the last step by expanding … Continue reading