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MAT425 Real Analysis L20

Prof. Thistleton

Series The idea of a series is ubiquitous in mathematics. In calculus II you defined the area under a curve in terms of a series. A basic tool in solving ordinary differential equations is to express a solution in terms of a series. Partial differential equations are often solved with Fourier series. The heart of many numerical solutions to partial differential equations is a technique called finite differences, which is an expression of a solution in terms of series. The list of these applications is very long.

To develop some intuition into series consider the following. Let a be a real number with |a| < 1. Define the sequence (a_n) as a_n = aⁿ. What is the limit $\lim_{n\rightarrow \infty} a_n$ ?

Define a new sequence, (s_n) as $s_n = a_1 + a_2 + \ldots + a_n$ . The terms of ( s_n ) are called partial sums for the obvious reason. What is the limit $\lim_{n\rightarrow \infty} s_n$ ?

The preceding example is quite famous and is called a geometric series. We will develop a mathematical framework for series. Fortunately, since a series is defined as a sequence of partial sums, much of our work is already done. Let (a_n) be a sequence. Define the n^th partial sum as $s_n = a_1 + a_2 + \ldots + a_n$ . We will try to find conditions under which $\lim_{n\rightarrow \infty} s_n$ is defined and, where possible, calculate $S \equiv \lim_{n\rightarrow \infty} s_n$ .

Definition 158

The sequence of partial sums, (s_n ), converges to S if, for every $\epsilon \gt 0$ there exists N such that $n \gt N \Rightarrow \vert s_n - S\vert < \epsilon$ .

Notice that we have really done nothing new here.

We present the following theorem in English.

Theorem 159

Whether or not a series converges isn't dependent upon starting place.
Whether or not a series converges isn't dependent upon a change in finitely many of its terms.

Theorem 163

Suppose the following series converge to the indicated limits.

$\begin{displaymath} \sum a_n = S\ \ \ \ \sum b_n = T\end{displaymath}$

What do you suppose is $\sum (a_n + b_n)$ ?
What do you suppose is $\sum c a_n$ ?

Series with Positive Terms We have already seen the monotone convergence theorem for sequences. Let (a_n) be a sequence with nonnegative terms. We have the following.

Theorem 166

$(s_n)\ bounded \Leftrightarrow \sum a_n$ converges.

We have already seen one example of a convergent series with positive terms, the geometric series. Another famous example is the harmonic series.

Result 168

Show that the harmonic series $\sum 1/n$ diverges.

The Cauchy Criterion What is a Cauchy sequence?

State and prove the Cauchy criterion for convergent series.

We have as an easy corollary: $\sum a_n$ converges only if $\lim_{n\rightarrow \infty} a_n = 0$ .Is the converse true?

The Comparison Test Suppose $\sum a_n$ and $\sum b_n$ are two series with nonnegative terms and suppose that $a_n \le b_n \ \forall \ n$ . Then $\sum b_n$ converges only if $\sum a_n$ converges.

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William Thistleton
11/10/1998