# indirect proof of identity theorem of power series

$\sum _{n=0}^{\mathrm{\infty}}}{a}_{n}{(z-{z}_{0})}^{n}={\displaystyle \sum _{n=0}^{\mathrm{\infty}}}{b}_{n}{(z-{z}_{0})}^{n$ | (1) |

is valid in the set of points $z$ presumed in the theorem (http://planetmath.org/IdentityTheoremOfPowerSeries) to be proved.

Antithesis: There are integers $n$ such that ${a}_{n}\ne {b}_{n}$; let $\nu $ ($\geqq 0$) be least of them.

We can choose from the point set an infinite sequence ${z}_{1},{z}_{2},{z}_{3},\mathrm{\dots}$ which converges to ${z}_{0}$ with ${z}_{n}\ne {z}_{0}$ for every $n$. Let $z$ in the equation (1) belong to $\{{z}_{1},{z}_{2},{z}_{3},\mathrm{\dots}\}$ and let’s divide both of (1) by ${(z-{z}_{0})}^{\nu}$ which is distinct from zero; we then have

$\underset{f(z)}{\underset{\u23df}{{a}_{\nu}+{a}_{\nu +1}(z-{z}_{0})+{a}_{\nu +2}{(z-{z}_{0})}^{2}+\mathrm{\dots}}}=\underset{g(z)}{\underset{\u23df}{{b}_{\nu}+{b}_{\nu +1}(z-{z}_{0})+{b}_{\nu +2}{(z-{z}_{0})}^{2}+\mathrm{\dots}}}$ | (2) |

Let here $z$ to tend ${z}_{0}$ along the points ${z}_{1},{z}_{2},{z}_{3},\mathrm{\dots}$, i.e. we take the limits ${lim}_{n\to \mathrm{\infty}}f({z}_{n})$ and ${lim}_{n\to \mathrm{\infty}}g({z}_{n})$. Because the sum of power series is always a continuous function^{}, we see that in (2),

$$\mathrm{left}\mathrm{side}\u27f6f({z}_{0})={a}_{\nu}\mathit{\hspace{1em}}\mathrm{and}\mathit{\hspace{1em}}\mathrm{right}\mathrm{side}\u27f6g({z}_{0})={b}_{\nu}$$ |

But all the time, the left and of (2) are equal, and thus also the limits. So we must have ${a}_{\nu}={b}_{\nu}$, contrary to the antithesis. We conclude that the antithesis is wrong. This settles the proof.

Note. I learned this proof from my venerable teacher, the number-theorist Kustaa Inkeri (1908–1997).

Title | indirect proof of identity theorem of power series |
---|---|

Canonical name | IndirectProofOfIdentityTheoremOfPowerSeries |

Date of creation | 2013-03-22 16:47:48 |

Last modified on | 2013-03-22 16:47:48 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Proof |

Classification | msc 40A30 |

Classification | msc 30B10 |