Proof of The Derived Series of Moduli of a Power Series Converges at Every Smaller Radius, Below a Radius of Convergence, and at All Radii

Routine facts about $\mathbb{R}$ and $\mathbb{N}_{0}$ are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity); the real sequences used below, built from given ones by shifting, reindexing and termwise operations, and from $k\mapsto k$ read in $\mathbb{R}$, exist by Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (ii), Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (iii), Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (v) and The Natural Numbers Read in ℝ: a Predicative Condition and a Function from ℕ₀ to ℝ (ii); and the results on series cited below are read in the current definition of series by The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (v). The idea is to compare the derived series at radius $\rho$ with the given series at the larger radius $\sigma$: the factor $k$ grows only linearly, while $(\rho/\sigma)^{k-1}$ decays geometrically, so their product stays bounded.

Step 1: a bound on $k\,(\rho/\sigma)^{k-1}$. We prove (i) in Steps 1 to 3, so let $0<\rho<\sigma$ and assume that $\sum_{k=1}^{\infty}|c_{k}|\,\sigma^{k-1}$ converges. Put $t=\sigma/\rho-1$, so that $t>0$ and $\sigma/\rho=1+t$, and put $m=\min(1,t)>0$. For $k\in\mathbb{N}$, Bernoulli's inequality with exponent $k-1\in\mathbb{N}_{0}$ (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (xii), applicable as $t\ge-1$) gives

$$\Bigl(\frac{\sigma}{\rho}\Bigr)^{k-1}=(1+t)^{k-1}\ge1+(k-1)\,t\ge m+(k-1)\,m=m\,k ,$$

since $1\ge m$ and $t\ge m$. Multiplying by the nonnegative number $(\rho/\sigma)^{k-1}$ (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (vi)) and using $(\rho/\sigma)^{k-1}(\sigma/\rho)^{k-1}=\bigl((\rho/\sigma)(\sigma/\rho)\bigr)^{k-1}=1$ (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii)), we get $m\,k\,(\rho/\sigma)^{k-1}\le1$, that is, with $M=1/m$,

$$k\,\Bigl(\frac{\rho}{\sigma}\Bigr)^{k-1}\le M\qquad(k\in\mathbb{N}).$$

Step 2: comparison of the terms. For every $k\in\mathbb{N}$ we have $\rho^{k-1}=(\rho/\sigma)^{k-1}\sigma^{k-1}$ and $\sigma^{k}=\sigma^{k-1}\sigma$ (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (ii), Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (i)), and all powers involved are nonnegative (Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality (vi)). Hence Step 1 gives

$$0\le k\,|c_{k+1}|\,\rho^{k-1}=k\Bigl(\frac{\rho}{\sigma}\Bigr)^{k-1}|c_{k+1}|\,\sigma^{k-1}\le M\,|c_{k+1}|\,\sigma^{k-1}=\frac{M}{\sigma}\,|c_{k+1}|\,\sigma^{k}.$$

Step 3: convergence. Let $a_{k}=|c_{k}|\,\sigma^{k-1}$ for $k\in\mathbb{N}$; by assumption $\sum_{k=1}^{\infty}a_{k}$ converges. The series $\sum_{k=1}^{\infty}|c_{k+1}|\,\sigma^{k}$ is the shifted series $\sum_{k=1}^{\infty}a_{1+k}$, which therefore converges (Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (iv), with $p=1$), and so does its multiple

$$\sum_{k=1}^{\infty}\frac{M}{\sigma}\,|c_{k+1}|\,\sigma^{k}$$

by the constant $M/\sigma$ (Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (i)). By Step 2 the terms of $\sum_{k=1}^{\infty}k\,|c_{k+1}|\,\rho^{k-1}$ are nonnegative and bounded by the terms of this convergent series, so the comparison test (Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (ii)) shows that the derived series converges. This proves (i).

Step 4: radii below $R$. For (ii), let $0<\rho<R$ and put $\sigma=(\rho+R)/2$, so that $0<\rho<\sigma<R$. By hypothesis $\sum_{k=1}^{\infty}|c_{k}|\,\sigma^{k-1}$ converges, and (i) applied to $\rho$ and $\sigma$ gives the convergence of $\sum_{k=1}^{\infty}k\,|c_{k+1}|\,\rho^{k-1}$.

Step 5: all radii. For (iii), let $\rho>0$ and put $\sigma=\rho+1$, so that $0<\rho<\sigma$. By hypothesis $\sum_{k=1}^{\infty}|c_{k}|\,\sigma^{k-1}$ converges, and (i) applied to $\rho$ and $\sigma$ again gives the convergence of $\sum_{k=1}^{\infty}k\,|c_{k+1}|\,\rho^{k-1}$.