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A convergent real power series with nonzero constant term has a convergent reciprocal power series on a smaller neighbourhood
Statement
Let have positive radius and . Then on some neighbourhood of , is represented by a convergent real power series about .
Facts & Assumptions
Given: The convergent power series with .
For , (For , , and for the series diverges).
A power series converges absolutely inside its radius, and its sum is continuous there (A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint, The sum of a real power series is continuous at every point strictly inside its interval of convergence).
Finite powers of a power series are represented by repeated Cauchy products (Inside the common radius the product of two power-series sums is represented by the Cauchy product of their coefficients).
Absolutely convergent double series may be regrouped without changing their sum (Fubini for double series: if converges then both iterated sums and the sum along every bijection converge to one and the same value).
Proof
Write , where . By absolute convergence, choose inside the radius so small that .
By [L1], for . Expand each power by [L3].
The total absolute sum of the expanded terms is bounded by . By [L4], regrouping by powers of gives a convergent reciprocal power series on the neighbourhood.
Depends on
- Inside the common radius the product of two power-series sums is represented by the Cauchy product of their coefficients
- For $|r| < 1$, $\sum_{k \ge 0} r^k = 1/(1-r)$, and for $|r| \ge 1$ the series diverges
- The sum of a real power series is continuous at every point strictly inside its interval of convergence
- A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint
- Fubini for double series: if $\sum_i \sum_j |a_{ij}|$ converges then both iterated sums and the sum along every bijection $\mathbb{N} \to \mathbb{N} \times \mathbb{N}$ converge to one and the same value
Used by
- A rational function with nonvanishing denominator is locally represented by geometric-series expansions Example
- The geometric series represents 1/(1-x) for |x|<1 and re-expands explicitly about every c with |c|<1 Example
- Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero Theorem
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Sources
- Power series, Encyclopedia of Mathematics (standard reference, not scraped)
- Northwestern Math 320-2 lecture notes (standard reference, not scraped)