Goal: Does a series converge or diverge? Lecture 24 – Divergence Test 1 Divergence Test (If a series converges, then sequence converges to 0.)

Example 1 – Converge/Diverge? 2 Example 2 – Converge/Diverge?

Example 3 – Converge/Diverge? 3 However,

Example 4 – Converge/Diverge? 4 However,

Goal: Does a series converge or diverge? Lecture 25 – Integral Test 5 Integral Test (The area under a function and infinite sum of the terms in a sequence defined by that function are related.)

6 If area under curve is bounded, then so is But then is a bounded, monotonic sequence. So it converges and thus converges. If area under curve is unbounded, then is also unbounded. And thus, diverges.

Example 1 – Converge/Diverge? 7 sequence converges to zero. No info from Divergence Test.

Example 2 – Converge/Diverge? 8 sequence converges to zero. No info from Divergence Test.

Example 3 – Converge/Diverge? 9 sequence converges to zero. No info from Divergence Test.

Example 4 – Converge/Diverge? 10 sequence converges to zero. No info from Divergence Test.

11 Lecture 26 – Ratio and Root Tests Goal: Does a series of positive terms converge or diverge? Ratio Test (Does ratio of successive terms approach some limit L? Then series is close to being geometric.)

12 Root Test (Does n th root of terms approach some limit L? Then series is close to being geometric.)

Example 1 – Converge/Diverge? 13 sequence converges to zero. No info from Divergence Test.

Example 2 14 For what values k does the series converge?

Example 3 – Converge/Diverge? 15 sequence converges to zero. No info from Divergence Test.

Example 4 – Converge/Diverge? 16

1. 2. Direct Comparison: 17 Lecture 27 – Comparison Tests

Limit Comparison: 18

Example 1 – Converge/Diverge? 19 sequence converges to zero. No info from Divergence Test.

Example 2 – Converge/Diverge? 20 sequence converges to zero. No info from Divergence Test.

Example 3 – Converge/Diverge? 21 sequence converges to zero. No info from Divergence Test.

Example 4 – Converge/Diverge? 22 sequence converges to zero. No info from Divergence Test.