MATH1012 - Lecture 17

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MATH1012 - Lecture 16
Skeleton Slides

Limit Laws for Sequences

• Limits of different sequences can be added, subtracted and multiplied by another limit or a scalar
  • Assuming that the limits are real numbers, not $\infty$
• See annotated slides for more details

Squeeze Theorem

• Squeeze theorem can be used to find the limit of oscillating sequences, such as ones containing a trigonometric function
• See Squeeze Theorem MATH1011

Squeeze Theorem

The sequence $\frac{\cos (n)}{n}$ is always enclosed by $\frac{1}{n}$ and $-\frac{1}{n}$

Bounded

• A sequence is bounded above/below if there is a value that $(a_n)$ does is always less/greater than
• A sequence is bounded if it is bounded both above and below
  • Any convergent sequence is bounded

Monotone Sequences

• Monotone (strictly) increasing: $a_1<a_2<a_3<a_4...$
• Monotone (strictly) decreasing: $a_1>a_2>a_3>a_4...$

Monotone Sequence Theorem

If sequence is monotone increase and bounded above, it converges
If sequence is monotone decreasing and bounded below, it converges

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MATH1012 - Lecture 18
Annotated Slides