Calculus II - Unit 2 - Part 1

What is a sequence?

A list of numbers written in a defined order a₁, a₂, a₃, a₄, . . .

When does a sequence converge?

A sequence converges if the limit as n goes to infinity equals a finite number.

When does a sequence diverge?

A sequence diverges if the limit as n goes to infinity is infinite.

When is sequence increasing?

It increases if a_n+1 ≥ a_n for all n ≥ 1.

When is sequence decreasing?

It decreases if a_n+1 ≤ a_n for all n ≥ 1.

When is a series monatomic?

It is monatomic when the series either decreases or increases not both.

When is a series bounded?

It is bounded when it is both bound above and below.

When is a series bounded below?

It is bounded below when the values ≥ the bound value.

When is a series bounded above?

It is bounded above when the values ≤ the bound value.

7 TRUE properties of sequences:

1. Every bounded, monotonic sequence is convergent.

2. Every increasing sequence is bounded below.

3. Every decreasing sequence is bounded above.

4. All convergent sequences are bounded.

5. Not all bounded sequences are convergent.

6. Not all convergent sequences are monotonic.

7. Not all monotonic sequences are convergent.

What is a sequence of partial sums?

It is a sequence created by adding the first N terms of a series one by one.