11.1 Sequences Notes

Objectives:

1. Find the limit of a sequence.

2. Limit Laws for Sequences

3. Monotonic and Bounded Sequences

Sequence

A sequence is an ordered list of numbers, typically written as , where is a positive integer. Sequences are fundamental objects in calculus and analysis, providing a framework for understanding limits, convergence, and divergence.

The Limit of a Sequence

The limit of a sequence describes the value that the terms of the sequence approach as becomes very large.

• Intuitive Definition of a limit of a sequence: A sequence has the limit . If we can make the terms as arbitrarily close to as we want just by taking the term and making it sufficiently large. The way we would write that is:

  or

• If such exists, the sequence is said to converge (it is convergent). Otherwise, it diverges (it is divergent).

Precise Definition of a Limit of a Sequence

A sequence has the limit . If, for every , there exists an integer such that

if then

Meaning that after some point, all terms of the sequence are within of .

The terms of a convergent sequence eventually stay within any horizontal band for sufficiently large

Properties of Convergent Sequences

• Theorem: If and for integer , then .

Limit Laws for Sequences

If and are convergent sequences and is a constant, then:

• Power Law: for and .

• Squeeze Theorem for Sequences: If and , then .

Finding the Limit of Sequence Examples from 11.2 Notes:

4. This sequence does not converge; it oscillates between 1-1.

Monotonic; Increasing and Decreasing Sequences

• Increasing Sequence: for all

• Decreasing Sequence: for all

• Monotonic Sequence: A sequence that is either increasing or decreasing.

Example:

1. Determine if is increasing, decreasing, or monotonic.

   The Solution: is increasing.

Monotonic Sequences; Bounded Above and/or Below

• Bounded Above: There exists such that for all .

• Bounded Below: There exists such that for all .

• Bounded Sequence: A sequence that is both bounded above and below.

Example:

1. Find the bounds for

   Solution: is bounded below by 0 but not above.

2. Find the bounds for

   Solution: for all .

Monotonic Sequence Theorem

• Theorem: Every bounded, monotonic sequence is CONVERGENT.

• Specifically:

  1. An increasing and bounded above sequence converges.

  2. A decreasing and bounded below sequence converges.

Examples that apply the Monotonic Sequence Theorem:

1. Does converge? Solution: Yes, .

2. Does converge? Solution: No, .

3. Does converge? Solution: No, it oscillates and does not approach a single value.

Additional info: The above notes include all major definitions, theorems, and examples relevant to the study of limits and properties of sequences, as typically covered in a first course on sequences and series in Calculus.