Proof that power series converge uniformly on smaller intervals inside the radius of convergence

Builds on Power series Radius of convergence Supremum Geometric series Weierstrass M-test

Proof

We dominate the terms by a convergent geometric series and apply the Weierstrass M-test.

Let $\sum a_n (x - c)^n$ be a power series with radius of convergence $R > 0$ . Let $A = \set{ t \ge 0 : (|a_n| t^n) \text{ is bounded} }$ , so that $R = \sup A$ .
Let $0 < r < R$ . Choose a real number $s$ with $r < s < R$ . For example, take $s = \frac{r + R}{2}$ if $R < \infty$ , and $s = r + 1$ if $R = \infty$ .
Since $s < R = \sup A$ , the number $s$ is not an upper bound of $A$ , by the definition of the supremum. So there is $t \in A$ with $t > s$ .
Since $t \in A$ , there is $C \ge 0$ with $|a_n| t^n \le C$ for all $n$ . Since $0 < s < t$ , also $|a_n| s^n \le |a_n| t^n \le C$ for all $n$ .
Let $q = \frac{r}{s}$ . Then $0 < q < 1$ .
Let $x \in [c - r, c + r]$ . Then $|x - c| \le r$ , so by step 4, $|a_n (x - c)^n| \le |a_n| r^n = |a_n| s^n q^n \le C q^n.$
Let $M_n = C q^n$ . Since $|q| < 1$ , the geometric series $\sum_{n=0}^{\infty} M_n$ converges.
By steps 6 and 7, the Weierstrass M-test applies to the functions $x \mapsto a_n (x - c)^n$ on $[c - r, c + r]$ .
So the power series converges absolutely and uniformly on $[c - r, c + r]$ .

Idea

Slightly beyond $r$ , at some $s < R$ , the terms $|a_n| s^n$ stay bounded. On $[c - r, c + r]$ , each term is then at most a constant times $(r/s)^n$ . These bounds form a convergent geometric series that does not depend on $x$ , so the M-test gives uniform convergence.

Reading path

To understand this page from the ground up, read these 123 pages in order. The foundations come first. The path takes the proofs Proof of the Weierstrass M-test, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of strong induction from ordinary induction, Proof of the squeeze theorem for sequences, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof that limits preserve non-strict inequalities, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that a decreasing sequence bounded below converges, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that a series of nonnegative terms converges iff its partial sums are bounded.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Integers object
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Ordered pair definition
Addition of natural numbers definition
Biconditional definition
The successor function is injective statement
Truth table concept
Zero is not a successor statement
Modus ponens concept
The rationals form a field statement
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Cartesian product definition
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Subset definition
Propositional logic concept
Mathematical proof concept
The rationals form an ordered field statement
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Partial sum definition
Power series definition
Binary relation definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Order on the reals definition
Tautology definition
Sum of a finite geometric series statement
Uniform convergence definition
Radius of convergence definition
Convergent series definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Strong induction statement
Logical equivalence definition
Uniform convergence on a set definition
Sum of a series definition
Series of functions definition
Absolute convergence definition
Preorder definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Well-ordering principle statement
Double negation law statement
Total order definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Epsilon-N definition of the limit of a sequence statement
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Squeeze theorem statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Proof that a decreasing sequence bounded below converges proof
Decreasing sequence bounded below converges statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Geometric series example
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Comparison test statement
Bolzano–Weierstrass theorem statement
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Cauchy criterion for sequences statement
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Cauchy criterion for series statement
Proof that absolutely convergent series converge proof
Absolutely convergent series converge statement
Proof of the Weierstrass M-test proof
Weierstrass M-test statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof

Where this leads

478 pages build on this one.

Tag	`A4K6K70RWW`
URI	`m2://A4K6K70RWW`
Kind	proof
Writing	W0 stub
MSC	30B10, 40A30
Proves	Power series converge uniformly on smaller discs inside the radius of convergence
Uses	Power series, Radius of convergence, Supremum, Geometric series, Weierstrass M-test
Changed	4 November 2026

Proof that power series converge uniformly on smaller intervals inside the radius of convergence

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