Lecture 1 - Monotone Convergence Theorem

MATH2400

Example

$$\begin{array}{rl}& \\ \sum _{n=1}^{\infty }\frac{1}{n}& =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\dots \\ & \ge 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{16}+\dots \\ & \text{The bottom series diverges, }\therefore \text{ the top series also diverges.}\end{array}$$

Another Exercise

Show that if ${\lim }_{n\to \infty }\mid x_n\mid =0$ then ${\lim }_{n\to \infty }{x}_n=0$ (what happens if 0 is replaced by some number $\mathbf{a}$.)

Integral Test

Integral Test

Suppose $f(x)=n$ is a decreasing function $\ge 0$. n Let $a_n=f(x)$ for all positive integers n ($\forall {\mathbb{N}}^{+}|n$)

Then ${\int }_1^{\infty }f(x)dx$ converges