♔Renascence_5

摘要： $$\Large\displaystyle \int_0^\infty \frac{\ln \left(1+\dfrac{\pi^2}{4x} \right)}{e^{\sqrt{x}} 1}\mathrm{d}x$$ $\Large\mathbf{Solution:}$ Step 1 Split 阅读全文

摘要： $$\Large\displaystyle \sum_{n=1}^\infty\frac{\Gamma\left(n+\dfrac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$$ $\Large\mathbf{Solution:}$ First, in view of Legr 阅读全文

摘要： $$\Large\displaystyle \sum_{n=1}^\infty ( 1)^n \frac{H_n}{2n+1}=\mathbf{G} \frac{\pi}{2}\ln(2)$$ $\Large\mathbf{Proof:}$ $\Large\mathbf{Method~One:}$ 阅读全文

摘要： $$\Large\displaystyle \sum_{n=0}^\infty \frac{1}{F_{2n+1}+1}=\frac{\sqrt5}{2}$$ $\Large\mathbf{Proof:}$ Let $\phi=\dfrac{1+\sqrt{5}}{2}$ denote the go 阅读全文