Sequences & Series

Vocab

100

The definitions of a sequence and a series

Sequence: an ordered list of numbers

Series: the sum of a sequence

200

What does it mean for a sequence  a_n  to be bounded?

The sequence  a_n  has a bound  B , where:

|a_n| ≤ B    for all n

300

What does it mean for a sequence to be monotonic?

The sequence is EITHER:

- Always Increasing 

or 

- Always Decreasing.

400

What does it mean for a sequence  a_n  to converge?

lim_n->∞ of a_n  exists

500

The Divergence Test is

A test for series divergence. For a sequence a_n : 

If lim_n->∞ a_n ≠ 0, then the series diverges

If lim_n->∞ a_n = 0, then we don't know (Inconclusive)

Concept

100

If a sequence converges, can its series diverge? If so, give an example

Yes. Examples: 

{2, 2, 2, 2, ...}

{1, 1.1, 1.11, 1.111, 1.111, ...}

200

If a series converges, can its sequence diverge? If so, give an example

No. Divergence test says all sequences who don't converge to 0 have corresponding series that diverge.

300

If a sequence is bounded and increasing, can the sequence diverge? If so, give an example.

No. "Every bounded, monotonic sequence converges" (theorem). 

400

If a sequence converges to 0, can its series diverge? If so, give an example.

Yes. Example: 1/n diverges

500

If a sequence has an upper bound and a lower bound, can the sequence diverge? If so, give an example.

Yes. Examples: (-1)^n, or sin(n)

()

200

How are sequences  a_n  like functions?

For the domain of positive integers  n , the range is the corresponding term of the sequence  a(n) .

They can be graphed.