Question

Prove the following result. Theorem. Let $\sum a_n$ be a series of real numbers that converges, but does not converge absolutely, and let $\alpha \le \beta$ be two given numbers in the extended reals. Then there exists a rearrangement $\sum a'_n$, with partial sums $s'_n$, such that: $\lim \inf s'_n = \alpha$, $\lim \sup s'_n = \beta$.

          Prove the following result. Theorem. Let $\sum a_n$ be a series of real numbers that converges, but does not converge absolutely, and let $\alpha \le \beta$ be two given numbers in the extended reals. Then there exists a rearrangement $\sum a'_n$, with partial sums $s'_n$, such that:
$\lim \inf s'_n = \alpha$, $\lim \sup s'_n = \beta$.

Prove the following result. Theorem. Let ∑ an be a series of real numbers that converges, but does not converge absolutely, and let α≤β be two given numbers in the extended reals. Then there exists a rearrangement ∑ a'n, with partial sums s'n, such that:
liminf s'n = α, limsup s'n = β.

Added by Jaime Z.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Linh Vu

04/11/2020

Step 1

We are also given two extended real numbers, α and β, with α ≤ β. Our goal is to prove that there exists a rearrangement of the series, denoted as E an', with partial sums sn', such that the limit inferior of sn' is α and the limit superior of sn' is β. Show more…

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Prove the following result. Theorem. Let sum a_(n) be a series of real numbers that converges, but does not converge absolutely, and let alpha <=eta be two given numbers in the extended reals. Then there exists a rearrangement sum a_(n)^('), with partial sums s_(n)^('), such that: liminfs_(n)^(')=alpha ,lim_()sups_(n)^(')=eta . Prove the following result. Theorem. Let E an be a series of real numbers that converges, but does not converge absolutely, and let a < be two given numbers in the extended reals. Then there exists a rearrangement Ean, with partial sums sn, such that: lim inf sn =, lim sup s=.