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The radius-one series with coefficients $1 / (n + 1)^{2}$ , $1 / (n + 1)$ and $1$ realise absolute, conditional and divergent endpoint behaviour

Statement

Each of

$\sum_{n \geq 0} \frac{x^{n}}{ι (n + 1)^{2}}, \sum_{n \geq 0} \frac{x^{n}}{ι (n + 1)}, \sum_{n \geq 0} x^{n}$

has radius $1$ . At $x = \pm 1$ the first converges absolutely; the second converges conditionally at $- 1$ and diverges at $1$ ; the third diverges at both endpoints.

Facts & Assumptions

Given: The three displayed real power series.

[L1]

The relevant coefficient roots tend to $1$ , because $ι (n)^{1 / n} \to 1$ and limits respect products and reciprocals ( $n^{1 / n} \to 1$ , The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Algebra of limits: sums, scalar multiples, products and quotients).

[L2]

Cauchy–Hadamard converts that limit into radius $1$ (Cauchy–Hadamard: the reciprocal radius is ${lim sup}_{k \to \infty} ∣ a_{k + 1} ∣^{1 / (k + 1)}$ , with the zero and infinite cases included).

[L3]

The $p$ -series converges for rational $p > 1$ and diverges for $p \leq 1$ , while the alternating harmonic series converges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ , The alternating series test: if $(b_{k})$ is nonincreasing with $b_{k} \to 0$ then $\sum_{k} (- 1)^{k} b_{k}$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_{n}$ ).

Verification

technique · cases

1.1

By [L1], all three Cauchy–Hadamard limit superiors equal $1$ , so [L2] gives radius $1$ in each case.

L1L2

1.2

For the squared-denominator series, absolute values at either endpoint form the $p$ -series with $p = 2$ , which converges by [L3].

assume-case squaredL3

1.3

For the first-power denominator, $x = 1$ gives the divergent harmonic series, while $x = - 1$ gives a convergent alternating series whose absolute series is harmonic.

assume-case harmonicL3

1.4

For the constant coefficients, at either endpoint the terms have absolute value $1$ and do not tend to zero, so both endpoint series diverge.

assume-case constantgiven

2.1

The coefficient families in steps 1.2--1.4 exhaust the displayed series and give the asserted absolute, conditional, and divergent endpoint behaviours.

step 1.1step 1.2step 1.3step 1.4cases-exhaustive∎

Depends on

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Sources

Power series, Encyclopedia of Mathematics (standard reference, not scraped)
MIT 18.100C, Lecture 11: Power Series (standard reference, not scraped)

The radius-one series with coefficients 1/(n+1)21/(n+1)2, 1/(n+1)1/(n+1) and 11 realise absolute, conditional and divergent endpoint behaviour