Section 10.4

Convergence & Divergence

How do we know if an infinite sum adds up to a finite number? In this section, we define convergence formally and introduce our first test: The Divergence Test.

Definition of Convergence

Recall that the value of a series is defined as the limit of its partial sums .

Convergent

If the limit of partial sums exists and is finite:

Divergent

If the limit does not exist or is infinite:

Note: Finding a formula for is often very difficult. Therefore, we usually rely on Convergence Tests to determine if the limit exists without finding the value itself.

The Divergence Test

If a series adds up to a finite number, the terms must eventually get exceedingly small (approach 0). If they don't, the sum will grow forever.

Theorem

If , then Diverges.

This is also called the "nth Term Test for Divergence".

WARNING: The Converse is False

The fact that does NOT guarantee convergence.

Passes Limit Check (Limit = 0)

But Diverges

(Harmonic Series)

The Divergence Test can only tell you if a series Diverges. It can never prove Convergence.

Absolute vs Conditional

Not all convergent series are created equal. Some converge "strongly" (even if you make all terms positive), while others only converge because their signs oscillate.

Absolute Convergence

The series of absolute values converges.

Conditional Convergence

The series converges, but the series of absolute values diverges.

Rearrangement

Riemann Rearrangement Theorem

We usually think . But for infinite series, the order of addition matters!

Absolutely Convergent Series

You CAN rearrange terms in any order. The sum will always be the same.

Conditionally Convergent Series

You CANNOT rearrange terms freely.

"If a series is conditionally convergent, its terms can be rearranged to sum to any real number, or even diverge to infinity."

Worked Examples

Example 1: Using the Divergence Test

Determine if converges or diverges.

Step 1: Check Limit of Terms

Take the limit of as :

Step 2: Apply Theorem

Since , the series Diverges by the Divergence Test.

Example 2: Inconclusive Test

Determine if converges using the Divergence Test.