Introduction to Sequences

Sequences

Think fundamentals today

Open Ended

Before we get into the mathematical definition of a sequence, I want you to tell me what a sequence is in your own words. You've got 1 minute. Go!

Sequences

• A list of numbers written in a definite order

• A sequence can be defined as a function whose domain is the set of positive integers

• We deal with infinite sequences

Sequence Notation

• {a₁, a₂, a₃, ...}

• {a_n}

• Sequences can be defined by giving a formula for the nth term.

• For example:  $a_n=\frac{n}{n+1}$  

• The n does not have to start at 1

Fill in the Blank

$$a_n=\sqrt{n-3},\ n\ge3.$$  What is the first term in the sequence?

Limit of a Sequence

A sequence  $\left\{a_n\right\}$   has the limit  $L$   and we write:
 $\lim_{n\rightarrow\infty}a_n=L$  or  $a_n\rightarrow L\ as\ n\rightarrow\infty$  

If the limit exists, the sequence converges (or is convergent).
Otherwise, the sequence diverges (or is divergent).

Different Topic, Same Concept

• Limit Laws hold for the limits of sequences

• The Squeeze Theorem can be adapted for sequences.

Multiple Choice

 Let's say there is a sequence where $$a_1<a_2<a_3$$ . What would you call this type of sequence?

• Increasing

• Decreasing

• Monotonic

• None of the above

Multiple Choice

 Let's say there is a sequence where $$a_1>a_2>a_3$$ . What would you call this type of sequence?

• Increasing

• Decreasing

• Monotonic

• None of the above

Fill in the Blank

What is a sequence called when it is either increasing or decreasing?

Bounded Sequences

• A sequence is considered bounded if it bounded both above AND below.

• For instance, the sequence $a_n=n$  is bounded below since  $a_n>0$ but not above.

• The sequence  $a_n=\frac{n}{n+1}$  is bounded because  $0<a_n<1$  for all  $n$  

Convergence

• Is every bounded sequence convergent?

• Is every monotonic sequence convergent?

• Turns out that a sequence is convergent if it is both bounded AND monotonic.