Section 10.2

More on Sequences

If a sequence is always increasing (Monotonic) but stuck below a ceiling (Bounded), it MUST converge. This is the Monotone Convergence Theorem.

Convergence Theorem

The Ceiling Trap

If you are always walking upstairs (Monotonic Increasing) but there is a roof above your head (Bounded), you must eventually hit the roof (Converge). You cannot go up forever without hitting infinity unless you stop at a limit.

Sequence increasing but trapped by y=1.

Theorem: Every Bounded, Monotonic sequence is Convergent.

Worked Examples

1. Show Monotonicity

. Is it increasing or decreasing?

Derivative Test

Analysis

For

, top is negative.
So the sequence is Decreasing.

2. Show Boundedness

Upper & Lower

Max value is at

(1).
All terms are positive (>0).

Result

Bounded in

3. Factorials

Is monotonic?

Ratio Test for Monotonicity

Compare