Convergence of Real Series

Convergence

Since a real series $\sum _{n\in \mathcal{D}}{a}_n$ is just a real sequence, the definitions and terminology for convergence of series is the same as those used for convergence of sequences. What is different, however, is the notation used.

Notation: Convergence of Real Series

If $\sum _{n\in \mathcal{D}}{a}_n$ converges to $L$, we write

$$\sum _{n\in \mathcal{D}}{a}_n=L$$

Instead of calling $L$ the limit of $\sum _{n\in \mathcal{D}}{a}_n$, we usually say it is its value.

Absolute Convergence

Definition: Absolute Convergence of Real Series

A real series $\sum _{n\in \mathcal{D}}{a}_n$ converges absolutely to $L$ iff the series $\sum _{n\in \mathcal{D}}|a_n|$ converges to $L$.

Properties

Theorem: Absolute Convergence $⟹$ Convergence

If a real series is absolutely convergent towards $L\in \mathbb{C}$, then it is also convergent towards $L$.

PROOF

TODO

Conditional Convergence

Definition: Conditional Convergence

A real series is conditionally convergent iff it is convergent but not absolutely convergent.

Properties

The Riemann Series Theorem

Let $(a_n{)}_{n\in \mathcal{D}}$ be an infinite real series.

Definition: Rearrangement

A rearrangement of $(a_n{)}_{n\in \mathcal{D}}$ is another infinite real series $(a_n^{\sigma }{)}_{n\in \mathcal{D}}$ which can be expressed as

$$a_n^{\sigma }=a_{\sigma (n)}$$

for some permutation $\sigma$ of $\mathcal{D}$.

For each $L\in \mathbb{R}$, there exists a rearrangement of $(a_n{)}_{n\in \mathcal{D}}$ which converges to $L$. There is exist also a rearrangement which diverges to $+\infty$, a rearrangement which diverges to $-\infty$ and a rearrangement which simply diverges.

INTUITION

The terms of every conditionally convergent real series can also be rearranged to produce a sum which is equal to any real number of our choice, or which diverges to infinity or negative infinite or just does not approach any limit, finite or infinite.

PROOF

TODO

Criteria for Convergence and Divergence

Theorem: Divergence Test

Let $\sum _{n\in \mathcal{D}}{a}_n$ be a real series.

If the sequence $(a_n{)}_{n\in \mathcal{D}}$ diverges or its limit is not zero, then $\sum _{n\in \mathcal{D}}{a}_n$ diverges.

PROOF

TODO

Theorem: Minorant Divergence Test

Let $\sum _{n\in \mathcal{D}}{a}_n$ be a real series.

If there exists some divergent real series $\sum _{n\in \mathcal{D}}{b}_n$ and some integer $N$ such that $0\le b_n\le a_n$ for all $n\ge N$, then $\sum _{n\in \mathcal{D}}{a}_n$ also diverges.

PROOF

TODO

Theorem: Majorant Convergence Test

Let $\sum _{n\in \mathcal{D}}{a}_n$ be a real series.

If there exists some convergent real series $\sum _{n\in \mathcal{D}}{b}_n$ and some integer $N$ such that $|a_n|\le b_n$ for all $n\ge N$, then $\sum _{n\in \mathcal{D}}{a}_n$ is absolutely convergent.

PROOF

TODO

Theorem: Quotient Test

Let $\sum _{n\in \mathcal{D}}{a}_n$ be a real series.

If the sequence $\left|\frac{a_{n+1}}{a_n}\right|$ converges, then $\sum _{n\in \mathcal{D}}{a}_n$ is:

• absolutely convergent if

$$\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|<1$$

• divergent if

$$\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|>1$$

NOTE

If $\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|=1$, then this theorem is useless.

PROOF

TODO

Theorem: Square Root Test

Let $\sum _{n\in \mathcal{D}}{a}_n$ be a real series.

If the sequence $\sqrt[n]{|a_n|}$ converges, then $\sum _{n\in \mathcal{D}}{a}_n$ is:

• absolutely convergent if

$$\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}<1$$

• divergent if

$$\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}>1$$

NOTE

If $\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}=1$, then this theorem is useless.

PROOF

TODO