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indirect proof of identity theorem of power series
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(Proof)
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(1) |
is valid in the set of points $z$ presumed in the theorem to be proved.
Antithesis: There are integers $n$ such that $a_n \neq b_n$ ; let $\nu$ ($\geqq 0$ ) be least of them.
We can choose from the point set an infinite sequence $z_1,\,z_2,\,z_3,\,\ldots$ which converges to $z_0$ with $z_n \neq z_0$ for every $n$ . Let $z$ in the equation (1) belong to $\{z_1,\,z_2,\,z_3,\,\ldots\}$ and let's divide both sides of (1) by $(z-z_0)^{\nu}$ which is distinct from zero; we then have
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(2) |
Let here $z$ to tend $z_0$ along the points $z_1,\,z_2,\,z_3,\,\ldots$ , i.e. we take the limits $\lim_{n\to\infty}f(z_n)$ and $\lim_{n\to\infty}g(z_n)$ . Because the sum of power series is always a continuous function, we see that in (2), $$\mathrm{left\,side\,}\longrightarrow f(z_0) = a_{\nu}\quad\mathrm{and}\quad \mathrm{right\,side\,}\longrightarrow g(z_0) = b_{\nu}$$ But all the time, the left and right side 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).
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"indirect proof of identity theorem of power series" is owned by pahio.
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| Keywords: |
proof by contradiction |
This object's parent.
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Cross-references: Kustaa Inkeri, proof, continuous function, power series, sum, limits, divide, equation, converges, sequence, infinite, integers, points, valid
This is version 6 of indirect proof of identity theorem of power series, born on 2007-03-05, modified 2009-01-06.
Object id is 9030, canonical name is IndirectProofOfIdentityTheoremOfPowerSeries.
Accessed 924 times total.
Classification:
| AMS MSC: | 30B10 (Functions of a complex variable :: Series expansions :: Power series ) | | | 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions) |
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Pending Errata and Addenda
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