総和総乗問題

\[\sum_{k=0}^{\infty}\frac{1}{k^{2}+1}=?\]

\[\lim_{n\rightarrow\infty}\frac{1}{\sqrt{n}}\sum_{k=0}^{n}\frac{\left(-1\right)^{k}}{B\left(n-k+1,k+1\right)\left(2k+1\right)}=?\]

\[\sum_{k=0}^{n}C\left(n,k\right)\sum_{j=k}^{n}C\left(n+1,j+1\right)=?\]

\[\frac{\zeta\left(2\right)}{\pi^{2}}-\frac{\zeta\left(4\right)}{\pi^{4}}+\frac{\zeta\left(6\right)}{\pi^{6}}-\frac{\zeta\left(8\right)}{\pi^{8}}+\cdots=?\]

\[\sum_{k=1}^{\infty}\frac{1}{\left(4k\right)^{3}-4k}=?\]

\[\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdots=?\]

\[\sum_{k=0}^{\infty}C\left(3n,3k\right)=?\]

\[\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\cdots+\frac{1}{\sqrt{28}+\sqrt{29}}+\frac{1}{\sqrt{29}+\sqrt{30}}=?\]

\[\sum_{k=0}^{n}kC^{2}\left(n,k\right)=?\]

\[\sum_{k=0}^{n}\frac{k!}{k!+\left(n-k\right)!}=?\]

\[\lim_{n\rightarrow\infty}\prod_{k=1}^{n}\left(1+\frac{k}{n^{2}}\right)=?\]

\[ \frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+\cdots+\frac{100}{98!+99!+100!}=? \]