% This is lecture no. 10 of MH3100/: Real Analysis I.

\chapter{Series}%
\label{cha:series}

\section{Series}%
\label{sec:series}

\begin{definition}
	Series.
	An (infinite) \textbf{series} of real numbers is a formal expression of the form
	\[
	a_1 + a_2+a_3+\dotsb
	\]
	where $\left\{a_n\right\}$ is a sequence or real numbers.
	The terms in the sequence $\left\{a_n\right\}$ are called \textbf{terms} of the series.
	We use the notation $\sum_{n=1}^{\infty} a_n$ (or simply $\sum a_n$) to denote the above series.
	For each $n\in\mathbb{N}$, the number $a_1+a_2+\dotsb+a_n$ is called the $n$-th \textbf{partial sum} of the series $\sum a_n$.
\end{definition}

% \begin{exmp}
% 	Series.
% 	\begin{enumerate}
% 		\item Geometric Series
% 			\[
% 			\sum_{n=0}^{\infty} ar^{n} = a + ar + ar^2 + ar^3 + ar^{4} + \dotsb
% 			.\]
% 		\item Harmonic Series
% 			\[
% 			\sum \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \dotsb
% 			.\]
% 		\item Alternating Harmonic Series
% 			\[
% 			\sum \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dotsb
% 			.\]
% 		\item $p$-series
% 			\[
% 			\sum \frac{1}{n^{p}} = 1 + \frac{1}{2^{p}} + \frac{1}{3^{p}} + \frac{1}{4^{p}} + \frac{1}{5^{p}} + \dotsb
% 			.\]
% 	\end{enumerate}
% \end{exmp}

\section{Convergent Series}%
\label{sec:convergent_series}

\begin{definition}
	Convergence of Series.
	A series $\sum a_n$ is said to be \textbf{convergent} if its sequence of partial sums $\left\{s_n\right\}$, where $s_n\defas a_1+a_2+\dotsb+a_n$, is convergent.
	In this case, the limit $A$ of the sequence of partial sums $\left\{s_n\right\}$ is called the \textbf{sum} of the series; and we write
	\[
	A = \sum_{n=1}^{\infty} a_n
	\qquad
	\left(\textbf{or simply }A = \sum a_n\right)
	\]
	to mean that $A$ is the sum of the series, and say that the series $\sum a_n$ \textbf{converges} to $A$.
	Otherwise the series is said to be \textbf{divergent}.
\end{definition}

\begin{thrm}
	Convergence of Terms of Convergent Series.
	If a series $\sum a_n$ is \textbf{convergent}, then the sequence of its \textbf{terms} $\left\{a_n\right\}$ \textbf{converges to} 0.
\end{thrm}

\begin{proof}
	For each $n\in\mathbb{N}$, both sequences of partial sums
	\[
	\left\{\sum_{i=1}^{n} a_i\right\}^{\infty}_{n=1}
	\quad
	\text{and}
	\quad
	\left\{\sum_{i=1}^{n-1} a_i\right\}^{\infty}_{n=1}
	\]
	converge to the sum of $\sum a_n$.
	Therefore, the Algebraic Limit Theorem asserts that the sequence
	\[
	\left\{\sum_{i=1}^{n} a_i-\sum_{i=1}^{n-1} a_i\right\}^{\infty}_{n=1}=\left\{a_n\right\}^{\infty}_{n=1}
	\]
	converges to 0.
\end{proof}

\begin{thrm}
	Algebraic Limit Theorem for Series.
	If the series $\sum a_n$ converges to $A$ and the series $\sum b_n$ converges to $B$, then
	\begin{enumerate}
		\item for each $c\in\mathbb{R}$, $\sum ca_n$ converges to $cA$; and
		\item $\sum(a_n+b_n)$ converges to $A+B$.
	\end{enumerate}
\end{thrm}

\begin{proof}
	For each $n\in\mathbb{N}$, let $s_n$ and $t_{n}$ denote the $n$-partial sums of, respectively, the series $\sum a_n$ and $\sum b_n$.
	Then $s_n\to A$ and $t_n\to B$.
	\begin{enumerate}
		\item The sequence of partial sums of the series $\sum ca_n$ is $\left\{cs_n\right\}$, which converges to $cA$ by the Algebraic Limit Theorem.
		\item The sequence of partial sums of the series $\sum (a_n + b_n)$ is $\left\{s_n+t_n\right\}$, which converges to $A+B$ by the Algebraic Limit Theorem.
	\end{enumerate}
	As required.
\end{proof}

\begin{thrm}
	Cauchy Criterion for Series.
	\begin{enumerate}
		\item A series $\sum a_n$ is \textbf{convergent} iff
			\[
			\forall\varepsilon>0,\exists N\in\mathbb{N}\suchthat\forall n\in\mathbb{N}\quad n>N\implies\abs{a_{N+1}+a_{N+2}+\dotsb+a_n}<\varepsilon
			.\]
		\item Equivalently, a series $\sum a_n$ is \textbf{convergent} iff
			\begin{multline*}
				\forall\varepsilon>0,\exists N\in\mathbb{N}\suchthat\forall n,m\in\mathbb{N}\\
				n>m>N\implies\abs{a_{m+1}+a_{m+2}+\dotsb+a_n}<\varepsilon.
			\end{multline*}
	\end{enumerate}
\end{thrm}

\begin{proof}
	This is a direct application of the Cauchy Criterion to the sequence of partial sums of the series.
\end{proof}
