Rearrangement diverges then original series also diverges?

Question: if $$sum y_n$$ is "any" rearrangement of series $$sum x_n$$ , where $$sum x_n$$ is series of positive terms. Then, if $$sum y_n$$ diverges then original series $$sum x_n$$ also diverges?

I think yes. Because series $$sum x_n$$ is series of positive terms and hence if we rearranged its terms then sum does not changed.

Mathematics Asked by Akash Patalwanshi on November 18, 2021

Assume that $$(x_n)_{ngeq 1}$$ is a sequence of real numbers and that either

$$sum_{n=1}^{infty} max{x_n, 0} < infty qquadtext{or}qquad sum_{n=1}^{infty} max{-x_n, 0} < infty$$

holds. (This in particular prevents the series $$sum_{n=1}^{infty} x_n$$ from converging conditionally.) Then we can prove that any rearrangement $$(y_n)_{ngeq1}$$ of $$(x_n)_{ngeq1}$$ satisfies

$$sum_{n=1}^{infty} x_n = sum_{n=1}^{infty} y_n,$$

regradless of whether they converge or not. (Note that each of these sums always has a value in the extended real number line $$[-infty,infty]=mathbb{R}cup{-infty,infty}$$, and then the equality holds in $$[-infty,infty]$$.)

It can be proved by focusing on OP's case (where $$x_n$$'s are non-negative). Indeed, assuming that $$x_n geq n$$ and $$sigma:mathbb{N}_1tomathbb{N}_1$$ is a bijection so that $$y_n=x_{sigma(n)}$$, then for each given $$m geq 1$$ there exists $$N$$ such that $${1,dots,m}subseteq{sigma(1),dots,sigma(N)}$$. Then for any $$ngeq N$$, we have

$$sum_{k=1}^{m} x_k leq sum_{k=1}^{n} y_{k} leq sum_{k=1}^{infty} x_k.$$

Letting $$ntoinfty$$, this shows that

$$sum_{k=1}^{m} x_k leq sum_{k=1}^{infty} y_{k} leq sum_{k=1}^{infty} x_k$$

for any $$m geq 1$$, then letting $$m to infty$$ proves the desired equality.

Answered by Sangchul Lee on November 18, 2021

A series of nonnegative numbers converges iff the series converges absolutely. A series converges absolutely iff every rearrangement of that series converges to the same limit.

Since there is a rearrangement that does not converge, the original series cannot converge absolutely. But the terms are nonnegative, so the original series diverges.

Answered by Andrew Shedlock on November 18, 2021

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