$$\frac{x}{{{e^x} - 1}} = \sum\limits_{k = 0}^{ + \infty } {\frac{{\operatorname{bernoulli} \left( k \right)}}{{k!}}{x^k},{\text{ }}\left| x \right| < 2\pi } $$

$$\frac{1}{2}x\frac{{{e^x} + 1}}{{{e^x} - 1}} = \sum\limits_{k = 0}^{ + \infty } {\frac{{\operatorname{bernoulli} \left( {2k} \right)}}{{\left( {2k} \right)!}}{x^{2k}}} ,{\text{ }}\left| x \right| < 2\pi $$

$$x\operatorname{ctg} x = \sum\limits_{n = 0}^{ + \infty } {{{\left( { - 1} \right)}^n}{B_{2n}}\frac{{{2^{2n}}}}{{\left( {2n} \right)!}}{x^{2n}},{\text{ }}\left| x \right| < \pi } $$

$$\operatorname{tg} x = \sum\limits_{n = 1}^{ + \infty } {\left| {{B_{2n}}} \right|\frac{{{2^{2n}} - 1}}{{\left( {2n} \right)!}}{x^{2n - 1}},{\text{ }}\left| x \right| < \pi /2} $$

$$\frac{1}{{\cosh \left( x \right)}} = \sum\limits_{n = 0}^{ + \infty } {\frac{{{\text{euler}}\left( n \right){x^n}}}{{n!}}} $$

$${x^2}{e^{{x^2}}} = \sum\limits_{k = 1}^{ + \infty } {\frac{{{x^{2k}}}}{{\left( {k - 1} \right)!}}} $$

$${e^{\frac{{{x^2}}}{2}}} = \sum\limits_{k = 0}^{ + \infty } {\frac{{{x^{2k}}}}{{\left( {2k} \right)!!}}} $$

$$\operatorname{erf} \left( x \right) = \frac{2}{{\sqrt \pi }}\int\limits_0^x {{e^{ - {t^2}}}dt} = \frac{2}{{\sqrt \pi }}\sum\limits_{n = 0}^{ + \infty } {\frac{{{{\left( { - 1} \right)}^n}{x^{2n + 1}}}}{{n!\left( {2n + 1} \right)}}} $$