The Basics of Infinite Series Review

We will now review some of the recent material regarding infinite series of real (and complex) numbers.

• On the Infinite Series of Real and Complex Numbers we formally defined an infinite series. If $(a_n)_{n=1}^{\infty}$ is a sequence of numbers, then the corresponding Infinite Series is denoted $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and is the sum of all terms in the sequence.

• The sequence $(a_n)_{n=1}^{\infty}$ is called the Sequence of Terms. Furthermore, we defined the Sequence of Partial Sums for this series, $(s_n)_{n=1}^{\infty}$, whose general term is given by:

(1)

\begin{align} \quad s_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + ... + a_n \end{align}

• On the Convergence and Divergence of Infinite Series page we said that a series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ Converges to a sum $s$ if the corresponding sequence of partial sums $(s_n)_{n=1}^{\infty}$ converges to $s$, i.e., $\displaystyle{\lim_{n \to \infty} s_n = s}$. A sequence that does not converge to any $s$ is said to Diverge.

• We then looked at some basic properties of infinite series on the Basic Properties of Convergent Infinite Series. We saw that if $\displaystyle{\sum_{n=1}^{\infty} a_n}$ and $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converge to $A$ and $B$ respectively, then:

(2)

\begin{align} \quad \sum_{n=1}^{\infty} [a_n + b_n] = A + B \end{align}

• Furthermore, if $c \in \mathbb{R}$, $c \neq 0$, then $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges to $A$ if and only if $\displaystyle{\sum_{n=1}^{\infty} ca_n}$ converges to $cA$.

• On the Rearrangements of Terms in Series of Real Numbers page we began to look at rearrangements of terms in series of real numbers. We said that if $(a_n)_{n=1}^{\infty}$ is a sequence of real numbers and if $f : \mathbb{N} \to \mathbb{N}$ is a bijection then a Rearrangement of the terms in the series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is a series $\displaystyle{\sum_{n=1}^{\infty} b_n}$ where for all $n \in \mathbb{N}$ we define:

(3)

\begin{align} \quad b_n = a_{f(n)} \end{align}

• On the Convergence of Rearranged Series of Real Numbers we quickly defined a series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ to be Absolutely Convergent if $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ converges, and if $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges and $\displaystyle{\sum_{n=1}^{\infty} \mid a_n \mid}$ does not converge, then the original series is said to be Conditionally Convergent.

• We then proved that if $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is an absolutely convergent series to a sum $A$ then any rearrangement $\displaystyle{\sum_{n=1}^{\infty} b_n}$ is also absolutely convergent to $A$.

• We also saw that if $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is a conditionally convergent series then for any $X$ and $Y$ such that $- \infty \leq Y \leq X \leq \infty$ there exists a rearrangement $\displaystyle{\sum_{n=1}^{\infty} b_n}$ such that if $(B_n)_{n=1}^{\infty}$ denotes the sequence of partial sums to this rearrangement, then $\displaystyle{\liminf_{n \to \infty} B_n = Y}$ and $\displaystyle{\limsup_{n \to \infty} B_n = X}$.

• On the Inserting and Removing Parentheses in Series of Real Numbers page we said that if $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is a series of real numbers and $p : \mathbb{N} \to \mathbb{N}$ is a strictly increasing function then if $\displaystyle{b_n = \sum_{k=p(n)+1}^{p(n+1)} a_k}$ (where $p(0) := 1$) then the series $\displaystyle{\sum_{n=1}^{\infty} b_n}$ is said to result from Adding Parentheses to $\displaystyle{\sum_{n=1}^{\infty} a_n}$:

(4)

\begin{align} \quad \sum_{n=1}^{\infty} a_n = \underbrace{(a_1 + a_2 + … + a_{p(1)})}_{b_1} + \underbrace{(a_{p(1) + 1} + a_{p(1) + 2} + … + a_{p(2)})}_{b_2} + … + \underbrace{(a_{p(n - 1) + 1} + a_{p(n - 1)+2} + … + a_{p(n)})}_{b_n} + ... \end{align}

• Furthermore, $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is said to result from Removing Parentheses from $\displaystyle{\sum_{n=1}^{\infty} b_n}$.

• We then saw that if $\displaystyle{\sum_{n=1}^{\infty} a_n}$ converges to $s$ and if $\displaystyle{\sum_{n=1}^{\infty} b_n}$ is a series obtained by adding parentheses to $\displaystyle{\sum_{n=1}^{\infty} a_n}$ then $\displaystyle{\sum_{n=1}^{\infty} b_n}$ converges to $s$.

• The converse to this result is not true in general.