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Tag Archives: Harmonic
Relation between harmonic numbers and Stirling numbers of the first kind
Prove that $$\left[n\atop 2\right] = H_{n1}\Gamma(n)$$ By induction on \( n\) we have for \( n=2\) $$\left[2\atop 2\right] = H_{1}\times\Gamma(1) = 1$$ Assume that $$\left[k\atop 2\right] = H_{k1}\Gamma(k)$$ Then by the recurrence relation $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading
Posted in Harmonic numbers, Striling numbers of first kind
Tagged Harmonic, number, numbers, proof, relation, stirling
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Relation between binomial sum and harmonic numbers
Prove that $$\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)^n1}{x} dx$$ By substitution we have $$\sum_{r=1}^n {n\choose r}\frac{(1)^{r+1}}{r}=\int^{1}_0 \frac{t^n1}{t1} dt = H_n$$ Note the last step by expanding … Continue reading