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The sum of a real power series is continuous at every point strictly inside its interval of convergence
Statement
If for , then is continuous at every satisfying .
Facts & Assumptions
Given: A power-series sum and a point strictly inside its radius.
The series converges uniformly on each closed interval strictly inside its radius (A power series converges absolutely and uniformly on every closed interval strictly inside its interval of convergence).
Every polynomial partial sum is continuous, since constants, the identity, powers, scalar multiples and finite sums are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).
A uniform limit of continuous real-valued functions is continuous (The uniform limit of continuous real-valued functions on a metric space is continuous).
Proof
Choose so small that lies strictly inside .
The polynomial partial sums are continuous on this interval by [L2] and converge uniformly there to by [L1].
By [L3], is continuous on that interval, and in particular at .
Depends on
- A power series converges absolutely and uniformly on every closed interval strictly inside its interval of convergence
- The uniform limit of continuous real-valued functions on a metric space is continuous
- Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
Used by
- The exponential is a continuous bijection from ℝ onto (0,∞) Corollary
- A convergent real power series with nonzero constant term has a convergent reciprocal power series on a smaller neighbourhood Lemma
- At a zero of a real-analytic function, either some first nonzero coefficient makes the zero isolated or every local coefficient vanishes Lemma
- Cosine has a smallest positive zero, lying strictly between zero and two Theorem
- The exponential function is strictly increasing Theorem
- Two real-analytic functions on an open interval that agree on a set with an accumulation point in that interval agree throughout the interval Theorem
Dependency tree · two levels
24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- MIT 18.100C, Lecture 11: Power Series (standard reference, not scraped)
- Power series, Encyclopedia of Mathematics (standard reference, not scraped)
- E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)