Lec 25 - Sequences & Series

- Increasing: Proof by induction.
- Bounded: Proof by induction.
- To find the limit, take the limit of the formula.

Recall:

Definition

Increasing, decreasing, monotonic, bounded sequence.

Aside: Completeness Axiom

If a set of real numbers has an upper bound then there is a least upper bound.

Theorem: Monotonic Sequence Theorem

Every bounded, monotonic sequence is convergent.

Proof:

By the axiom above, if it is bounded there is a least upper bound.

This implies

Example

Proof:

Play with the sequence

…

Taking the limit,

Therefore,

Series 11.2

Definition

An infinite series is an infinite sum of a sequence,

Definition

Definition

Definition

The geometric series is the sum of the geometric sequence.

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