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# boundedness of terms of power series

Theorem. If the set

$\{a_{0},\;a_{1}c,\;a_{2}c^{2},\;\ldots\}$ |

of the terms of a power series

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

at the point $z=c$ is bounded, then the power series converges, even absolutely, for any value $z$ which satisfies

$|z|<|c|.$ |

*Proof.* By the assumption, there exists a positive number $M$ such that

$|a_{n}c^{n}|<M\quad\forall\,n\,=\,0,\,1,\,2,\,\ldots$ |

Thus one gets for the coefficients of the series the estimation

$|a_{n}|<\frac{M}{|c|^{n}}.$ |

If now $|z|<|c|$, one has

$|a_{n}z^{n}|<M\left|\frac{z}{c}\right|^{n},$ |

and since the geometric series $\displaystyle\sum_{{n=0}}^{\infty}\left|\frac{z}{c}\right|^{n}$ is convergent, then also the real series $\displaystyle\sum_{{n=0}}^{\infty}|a_{n}z^{n}|$ converges.

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## Mathematics Subject Classification

40A30*no label found*30B10

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