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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Inside the common radius the product of two power-series sums is represented by the Cauchy product of their coefficients
Statement
Suppose and have radii . For ,
and the displayed product series converges absolutely.
Facts & Assumptions
Given: The two power series in the statement and a point in their common open radius.
Both numerical series converge absolutely there (A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint).
The Cauchy product of two absolutely convergent series converges absolutely to the product of their sums (The Cauchy product of two series: , If and both converge absolutely then their Cauchy product converges absolutely, with sum ).
Proof
Apply [L2] to the numerical series with terms and , whose absolute convergence is [L1].
Its th Cauchy-product term is , which gives the formula and absolute convergence.
Depends on
- A real power series converges absolutely inside its radius and diverges outside it, while either behaviour may occur at an endpoint
- The Cauchy product of two series: $c_n = \sum_{k=0}^{n} a_k b_{n-k}$
- If $\sum a_k$ and $\sum b_k$ both converge absolutely then their Cauchy product converges absolutely, with sum $AB$
Used by
- A convergent real power series with nonzero constant term has a convergent reciprocal power series on a smaller neighbourhood Lemma
- Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero Theorem
- The exponential addition formula exp(x+y)=exp(x)exp(y) Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 64 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Power series, Encyclopedia of Mathematics (standard reference, not scraped)
- E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)