# Tag Archives: Harmonic

Prove that $$\left[n\atop 2\right] = H_{n-1}\Gamma(n)$$ By induction on $n$ we have for $n=2$ $$\left[2\atop 2\right] = H_{1}\times\Gamma(1) = 1$$ Assume that $$\left[k\atop 2\right] = H_{k-1}\Gamma(k)$$ Then by the recurrence relation $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading | Tagged , , , , , | Leave a comment ## Relation between binomial sum and harmonic numbers Prove that$$\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

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