00:00 Intro

00:17 Meaning of reordering of series

01:13 Example: Reordering can change the value

02:32 2nd Example with a convergent series

04:58 Definition Reordering

06:10 Theorem for abs. convergent series

06:39 Proof of the Theorem

12:22 Credits

Q1: Is the series $\displaystyle \sum_{k = 1}^\infty (-1)^{k+1} \frac{1}{k}$ absolutely convergent?

A1: Yes, it is.

A2: No, it isn’t.

Q2: When do we call a series $\displaystyle \sum_{k = 1}^\infty a_{\tau(k)}$ a reordering of the series $\displaystyle \sum_{k = 1}^\infty a_k$?

A1: It’s called a reordering if $\tau : \mathbb{N} \rightarrow \mathbb{N}$ is map.

A2: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a injective map.

A3: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a surjective map.

A4: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a bijective map.

A5: It’s called a reordering if $\tau :\mathbb{N} \rightarrow \mathbb{N}$ is a bijective map with $\tau(1) = 1$.

Q3: Can one change the value of a convergent series with a reordering.

A1: Yes, that is always possible.

A2: No, this is never possible.

A3: Yes, this is possible but only if the series is not absolutely convergent.

Q4: Let $q \in \mathbb{R}$ with $|q| < 1$. Can one change the limit of the series $\displaystyle \sum_{k = 0}^\infty q^k$ with a reordering?

A1: Yes, this is always possible.

A2: No, this is not possible.

A3: It possible for $q < 0$ but not for $q \geq 0$.