Math 2.0

statement theorem

Power series converge uniformly on smaller discs inside the radius of convergence

Statement

Let ∑n=0∞an(x−c)n\sum_{n=0}^{\infty} a_n (x - c)^n be a power series with radius of convergence R>0R > 0. Then for every rr with 0<r<R0 < r < R, the series converges absolutely and uniformly on [c−r,c+r][c - r, c + r].

Remarks

Choose ss with r<s<Rr < s < R. The series converges at c+sc + s, so ∣an∣sn≤C|a_n| s^n \le C for some CC. The M-test with Mn=C(r/s)nM_n = C (r/s)^n applies. So the sum is continuous on (c−R,c+R)(c - R, c + R). The convergence need not be uniform on the whole interval: ∑xn\sum x^n is not uniformly convergent on (−1,1)(-1, 1). Mathlib states the result as HasFPowerSeriesOnBall.tendstoUniformlyOn.

Explanations

L1 human

Inside the interval of convergence, a power series converges uniformly once we stay away from the endpoints. Near the endpoints, the convergence can get slower and slower.

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To read this node, read these 27 nodes in this order. The foundations come first, and the list ends with this node.

Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementPointwise convergence, definition, no evidence levelPointwise convergencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionUniform convergence, definition, no evidence levelUniform convergencedefinitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionSeries of functions, definition, no evidence levelSeries of functionsdefinitionComparison test, statement, no evidence levelComparison teststatementUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionWeierstrass M-test, statement, no evidence levelWeierstrass M-teststatementPower series converge uniformly on smaller discs inside the radius of convergence, statement, no evidence levelPower series converge…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Sequence definition
  2. Real sequence definition
  3. Subsequence definition
  4. Bounded sequence definition
  5. Cauchy sequence definition
  6. Limit of a sequence definition
  7. Monotone sequence definition
  8. Series definition
  9. Cauchy sequences are bounded statement
  10. Monotone convergence theorem for sequences statement
  11. Convergent sequence definition
  12. Every sequence has a monotone subsequence statement
  13. Pointwise convergence definition
  14. Partial sum definition
  15. Power series definition
  16. Bolzano–Weierstrass theorem statement
  17. Radius of convergence definition
  18. Convergent series definition
  19. Uniform convergence definition
  20. Cauchy criterion for sequences statement
  21. A series of nonnegative terms converges iff its partial sums are bounded statement
  22. Uniform convergence on a set definition
  23. Series of functions definition
  24. Comparison test statement
  25. Uniformly Cauchy sequence of functions definition
  26. Weierstrass M-test statement
  27. Power series converge uniformly on smaller discs inside the radius of convergence statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Radius_of_convergence, Definition

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