Research proof index

For every integer $n\ge0$, the corrected harmonic approximation $H_{n+1}-\log(n+1)-\frac{1}{2(n+1)}$ is a lower bound for the Euler–Mascheroni constant $\gamma$.

Let $\gamma_3(n)=H_n-\log n-\frac{1}{2n}+\frac{1}{12n^2}-\frac{1}{120n^4},$ where $H_n$ is the $n$th harmonic number. Then $\gamma_3(n)<\gamma_3(n+1)$ for every integer $n\ge1$.

Let $\gamma_3(n)=H_n-\log n-\frac{1}{2n}+\frac{1}{12n^2}-\frac{1}{120n^4}.$ For every integer $n\ge1$, this corrected harmonic approximation is a strict lower bound for the Euler–Mascheroni constant: $\gamma_3(n)<\gamma.$