MATH1012 - Lecture 20

MATH1012 - Lecture 19
Skeleton Slides

Suppose that $0 \leq a_{n} \leq b_{n}$ for all sufficiently large $n$ , then:
- If $\sum_{n = 1}^{\infty} a_{n}$ is divergent then $\sum_{n = 1}^{\infty} b_{n}$ is divergent
- If $\sum_{n = 1}^{\infty} b_{n}$ is convergent then $\sum_{n = 1}^{\infty} a_{n}$ is convergent
In short:
- Larger than divergent > divergent
- Smaller than convergent > convergent
This test only applies for series with non-negative terms
- Example applications in lecture recording

For $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ with $a_{n} \geq 0$ and $b_{n} > 0$ , let: $lim_{n \to \infty} \frac{a _{n}}{b _{n}} = c$
- Remember Limit Laws!
We are given three cases:
- $c = 0$
  - If $\sum b_{n}$ converges so does $\sum a_{n}$
- $0 < c < \infty$
  - If one series converges, then so does the other
- $c = \infty$ (limit diverges to $\infty$ )
  - If $\sum a_{n}$ converges so does $\sum b_{n}$

A series where the terms alternate in sign are called alternating series
- e.g. $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots$
These are usually viewed as taking the alternating sum of $a_{n}$ : $\sum_{n = 1}^{\infty} (- 1)^{n - 1} a_{n}$

Alternating series converge when $a_{n}$ meets these conditions:
- $a_{n} \geq 0$
- $n \to \infty lim a_{n} = 0$
- $a_{n + 1} \leq a_{n}$ (eventually)

MATH1012 - Lecture 21
Annotated Slides

Quartz 5