Power Series

$\sum_{n = 0}^{\infty } n^{2} z^{n}$

$\sum_{n = 0}^{\infty } 3^{n}(z+1)^{n}$

$\sum_{n = 0}^{\infty }\left(\frac{n+1}{2 n+3}\right)^{n} z^{n}$;

$\sum_{n = 0}^{\infty } \frac{(n+1)(z-2)^{n}}{4^{n+2}}$

$\sum_{n = 0}^{\infty } n^{n} z^{n}$

$\sum_{n = 0}^{\infty }\left(\frac{n+2}{n+5}\right)^{n^{2}} z^{n}$.

$\sum_{n = 1}^{\infty } 2^{\ln^{2} n}(z-3)^{n}$;

$\sum_{n = 1}^{\infty }\left(\frac{z}{\sqrt{n}}\right)^{n}$;

$\sum_{n = 1}^{\infty } n !\left(\frac{z}{n}\right)^{n}$

$\sum_{n = 1}^{\infty } \frac{\sqrt{n^{n}}}{n !} z^{n}$;

$\sum_{n = 1}^{\infty } \frac{2^{n} z^{4 n}}{n^{2}}$;

$\sum_{n = 1}^{\infty } \frac{(n+1)^{2}}{3^{n}} z^{5 n}$.

$\sum_{n = 1}^{\infty } \frac{(n !)^{2}}{(2 n) !} z^{n}$;

$\sum_{n = 1}^{\infty } \frac{z^{n}}{(n !)^{\alpha }}, \alpha > 0$;

$\sum_{n = 1}^{\infty } n ! e^{-n^{\alpha }} z^{n}, \alpha > 1$;

$\sum_{n = 1}^{\infty } \frac{(k n) !}{n !(n+1) ! \ldots (n+k-1) !}, k \in N$;

$\sum_{n = 1}^{\infty } n ! z^{n !}$

$\sum_{n = 1}^{\infty }(2-\sqrt{e})(2-\sqrt[3]{e}) \ldots (2-\sqrt[n]{e}) z^{n}$.

$\sum_{n = 1}^{\infty } n\left(\cos \frac{1}{n}\right)^{2 n^{3}} z^{n}$;

$\sum_{n = 1}^{\infty } \frac{2^{n} \cos (3 \pi n / 4)}{n} z^{n}$;

$\sum_{n = 1}^{\infty }\left(\left(1-\frac{1}{n}\right)^{n} \operatorname {arctg} e^{-n}\right)(z+1)^{n}$;

$\sum_{n = 1}^{\infty }\left(\frac{n^{2}+3}{n^{2}+5}\right)^{n^{3}}(z-1)^{n}$;

$\sum_{n = 1}^{\infty }\left(1+\operatorname {arctg} \frac{n+1}{n^{2}}\right)^{n}(z+2)^{2}$;

$\sum_{n = 1}^{\infty }\left(\ln \cos \frac{1}{3^{n}}\right) z^{n}$.

$\sum_{n = 1}^{\infty } \frac{2^{n}(1+i)^{3 n}}{(n+1)(n+2)} z^{n}$;

$\sum_{n = 1}^{\infty }\left(\frac{1-i}{2}\right)^{n} z^{2 n}$;

$\sum_{n = 1}^{\infty } \frac{(2+i \sqrt{5})^{n}}{(3-i \sqrt{7})^{2 n}}(z-i)^{3 n}$;

$\sum_{n = 1}^{\infty } \frac{n ! z^{n}}{(1+i)(1+2 i) \ldots (1+n i)}$.

$\sum_{n = 1}^{\infty }(\sin (\sqrt{n+1}-\sqrt{n}))(x+1)^{n}$;

$\sum_{n = 1}^{\infty }\left(\sin \frac{n^{2}}{2^{n}}\right)(x-3)^{n}$;

$\sum_{n = 1}^{\infty } \frac{2^{n} n}{n^{n}}(x-1)^{2 n}$;

$\sum_{n = 1}^{\infty }\left(\arcsin \frac{1}{3^{n}}\right)(x-3)^{n}$

$\sum_{n = 1}^{\infty }\left(\operatorname {arctg} \frac{n^{2}+3}{n^{2} \sqrt{3}+1}\right)^{-n}\left(x-\frac{\pi }{3}\right)^{n}$;

$\sum_{n = 1}^{\infty } \frac{1}{n !}\left(\frac{n x}{e}\right)^{n}$.

$\sum_{n = 1}^{\infty } \frac{(x-1)^{n}}{n \sqrt{n}}.$;

$\sum_{n = 1}^{\infty }\left(\frac{2 n-1}{3 n+2}\right)^{n}(x+2)^{n}$;

$\sum_{n = 0}^{\infty } \frac{(-1)^{n} x^{n}}{2 n+1}$

$\sum_{n = 0}^{\infty } \frac{(-1)^{n-1} 2 n+3}{3 n^{2}+4} x^{2 n+1}$;

$\sum_{n = 1}^{\infty } \frac{1}{\sqrt[3]{n}}\left(\frac{x-1}{3}\right)^{n}$;

$\sum_{n = 1}^{\infty } \sqrt{\frac{n^{4}+3}{n^{3}+4 n}}(x+2)^{n}$.

$\sum_{n = 1}^{\infty } \frac{5^{n}+(-3)^{n}}{n+1} x^{n}$

$\sum_{n = 1}^{\infty } \frac{(x+1)^{n}}{\sqrt{n+1}} \ln \frac{3 n-2}{3 n+2}$;

$\sum_{n = 1}^{\infty } \frac{\sqrt[3]{2 n+1}-\sqrt[3]{2 n-1}}{\sqrt{n}}(x+3)^{n}$;

$\sum_{n = 1}^{\infty }(\sqrt[n]{a}-1) x^{n}, a > 0$;

$\sum_{n = 0}^{\infty } 3^{n}\left(n^{3}+2\right)(x-1)^{2 n}$;

$\sum_{n = 1}^{\infty } 4^{n^{2}}(x+1)^{n^{2}}$.

$\sum_{n = 1}^{\infty } \frac{(2 n-1) ! !}{n !}(x+2)^{n}$;

$\sum_{n = 1}^{\infty } \frac{(x+3)^{n^{2}}}{n^{n}}$

$\sum_{n = 1}^{\infty }\left(1-\frac{1}{n}\right)^{n^{2}} x^{n}$;

$\sum_{n = 1}^{\infty }\left(1+\frac{1}{2}+\ldots +\frac{1}{n}\right)(x-1)^{n}$;

$\sum_{n = 1}^{\infty } \frac{\left(2+(-1)^{n}\right)^{n}}{n}(x+1)^{n}$;

$\sum_{n = 1}^{\infty } \frac{(1+2 \cos (\pi n / 4))^{n} x^{n}}{\ln^{2}(n+1)}$.

$\sum_{n = 1}^{\infty } \frac{(x-1)^{n}}{n^{\alpha }}$;

$\sum_{n = 1}^{\infty }\left(\frac{(2 n-1) ! !}{(2 n) ! !}\right)^{\alpha } x^{n}$;

$\sum_{n = 1}^{\infty }\left(\frac{a^{n}}{n}+\frac{b^{n}}{n^{2}}\right) x^{n}, a > 0, b > 0$;

$\sum_{n = 1}^{\infty } \frac{x^{n}}{a^{n}+b^{n}}, \quad a > 0, \quad b > 0$;

$\sum_{n = 1}^{\infty } \frac{a(a-1) \ldots (a-(n-1))}{n !} \cdot x^{n}$;

$\sum_{n = 1}^{\infty }(-1)^{n-1}\left(\frac{2^{n}(n !)^{2}}{(2 n+1) !}\right)^{\alpha }(x-1)^{n}$.

$\sum_{n = 1}^{\infty } \frac{3^{-\sqrt{n}}}{\sqrt{n^{2}+n+1}}(x-1)^{n}$;

$\sum_{n = 0}^{\infty } \frac{(-1)^{[\sqrt{n}]}}{n} x^{n}$;

$\sum_{n = 1}^{\infty }\left(\frac{x}{\sin n}\right)^{n}$;

$\sum_{n = 1}^{\infty } \frac{(-1)^{n-1}}{n !}\left(\frac{n}{e}\right)^{n}(x-1)^{n}$;

$\sum_{n = 1}^{\infty } \frac{10^{\nu (n)}}{n}(1-x)^{n}, \quad \nu (n)$ is the number of digits of the number $n$.

$\sum_{n = 1}^{\infty } 2^{n} x^{n^{2}}$

$\sum_{n = 1}^{\infty } \frac{(x-1)^{n}}{a^{\sqrt{n}}}, a > 0 ;$

$\sum_{n = 1}^{\infty } \frac{n !}{a^{n^{2}}} x^{n}, a > 1$.

$\sum_{n = 1}^{\infty }(-1)^{n+1} \frac{\ln^{n} x}{3^{n}(n+1)}$;

$\sum_{n = 1}^{\infty } \frac{1}{2 n+3}\left(\frac{1-x}{1+x}\right)^{n}$;

$\sum_{n = 1}^{\infty } \frac{1}{(x-1)^{n}} \operatorname {tg} \frac{\pi }{2^{n}}$;

$\sum_{n = 1}^{\infty }\left(1+\frac{1}{n}\right)^{-n^{2}} e^{-n x}$;

$\sum_{n = 1}^{\infty } 2^{n} \cos^{n} x$;

$\sum_{n = 1}^{\infty } \frac{3^{3 n}(n !)^{3}}{(3 n) !} \operatorname {tg}^{n} x$.

$\sum_{n = 0}^{\infty } c_{n}^{k} z^{n}, \quad k \in N$

$\sum_{n = 0}^{\infty } c_{n} z^{k n}, \quad k \in N$

$\sum_{n = 0}^{\infty } \frac{c_{n}}{1+\left|c_{n}\right|} z^{n}$.

$\sum_{n = 0}^{\infty }\left(a_{n}+b_{n}\right) x^{n}$

$\sum_{n = 0}^{\infty } a_{n} b_{n} x^{n}$

for the $n$-th remainder of this series the asymptotic formula holds

i.e.

the coefficients $c_{n}$ of the power series are determined uniquely (uniqueness of the expansion of a regular function into a power series).

$f(z) = \frac{1}{z^{2}+a^{2}}, \quad a \neq 0$;

$f(z) = \frac{1}{z^{m}+a}, \quad a \neq 0, \quad m \in N$;

$z(x) = \frac{1}{3 z^{4}+10 z^{2}+3}$;

$z(x) = \frac{1}{2 z^{3}-6 z^{2}+z-3}$.

$P(x) = x^{2}-x-2, x_{0} = 0$

$P(x) = x^{2}+1, x_{0} = -1$;

$P(x) = x^{2}-x-6, x_{0} = 1$;

$P(x) = x^{2}-x+1, x_{0} = 2$;

$P(x) = 2 x^{2}+3 x-2, x_{0} = -1 / 2$;

$P(x) = x^{3}-8, x_{0} = -3$.

for all $x \in (0 ; 1)$ the series $\sum_{n = 0}^{\infty } a_{n} x^{n}$ converges;

for all $x$ such that $\left|x\right| < \min (1 ; R)$, where $R$ is the radius of convergence of the series $\sum_{n = 0}^{\infty } a_{n} x^{n}$, the equalities hold

if the sum $f(x)$ of this series is bounded on the interval $[0 ; 1)$, then the series $\sum_{n = 0}^{\infty } a_{n}$ converges and $\lim_{x \rightarrow 1-0} f(x) = \sum_{n = 0}^{\infty } a_{n}$;

if the series $\sum_{n = 1}^{\infty } a_{n}$ diverges, then $\lim_{x \rightarrow 1-0} f(x) = +\infty$.

$\sum_{n = 0}^{\infty }(-1)^{n}$;

$\frac{1}{2}+\sum_{n = 1}^{\infty } \cos n \theta , \quad -\pi \leqslant \theta \leqslant \pi ;$

$\sum_{n = 1}^{\infty } \sin n \theta , \quad -\pi \leqslant \theta \leqslant \pi$.

if $\lim_{n \rightarrow \infty } \frac{a_{n}}{b_{n}} = A$ and the series $\sum_{n = 1}^{\infty } a_{n}$ diverges, then

if $\lim_{n \rightarrow \infty } \frac{A_{n}}{B_{n}} = A$, where $A_{n} = \sum_{k = 1}^{n} a_{k}, B_{n} = \sum_{k = 1}^{n} b_{k}$, and if the series $\sum_{n = 1}^{\infty } A_{n}$ diverges, then