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\(\sum_{n=1}^{\infty} \frac{n^{k-1}}{e^{2\pi n}-1} = \int_0^{\infty} \frac{x^{k-1}}{e^{2\pi x}-1}\ dx\)

\(\sum_{n=1}^{\infty} \frac{1}{n^n} = \int_0^{1} \frac{1}{x^x}\ dx\)

\(k>2\mbox{, } k\mbox{ even}\)

\(\frac{B_k}{2k}\)

\(B_k\)

\(\sum_{n=0}^{\infty}\frac{1}{n!} = e \mbox{ en } \sum_{n=0}^{\infty}\frac{(-1)^n}{n!} = \frac{1}{e}\)

\(\sum_{n=1}^{\infty} \frac{1}{n^n} = \int_0^{1} \frac{1}{x^x}\ dx \mbox{ en } \sum_{n=1}^{\infty} \frac{(-1)^n}{n^n} = -\int_0^{1} x^x\ dx\)