Warm Up

n n 1

nn n

n

1. Which of the f ollowing sequences converge?

2 2 n!I . A I I . A

n 2 n 1 ! 2

2n 3I I I . A

5n 1(A) I only (B) I I only (C) I I I only

(D) I I and I I I only (E) I , I I , and

I I I

2. Consider the series:

a)What is the sum of the series?

b)How many terms are required in the partial sum to approximate the sum of the infinite series within 0.001?

1

1

14

3

n

n

Ratio and Root Tests for convergence/divergence

Let be a series with nonzero terms.

Find the ratio:

1. If the limit is < 1, the series converges.

2. If the limit is > 1 (including infinity), the series diverges.

3. If the limit = 1, the test is inconclusive. Choose a different convergence test

na1lim n

nn

a

a

The Ratio Test (Use for series with nn , n!, exponentials, etc.)

Example:

12 1

n

nn

Example:

1n!nen

Determine the convergence or divergence of:

0

2

!

n

n n

2 1

0

2

3

n

nn

n

1. 2.

0

11

n

n

n

n

0 !

n

n

n

n

3. 4.

The root test (use if every thing is raised to the nth power)

Let be a series with nonzero terms.naFind: lim n

nn

a

1. If limit is < 1, converges.

2. If limit is > 1 (including infinity), diverges.

3. If limit = 1, inconclusive. Choose a different test.

Examples:

1. 2

1

n

nn

e

n

1

1

2 1

n

n

n

n

2.

Guidelines for Convergence/Divergence.

1. Does the nth term approach 0? If not, the series diverges.

2. Is the series one of those special types – geometric, p-series, telescoping or alternating?

3. Can the integral test, root test, or ratio test be applied?

4. Can the series be compared directly or using a limit?

Putting it all together:Determine the convergence or divergence of the series below. If possible, find what the series converges to.

1

1

3 1n

n

n

1 6

n

n

1

!

10nn

n

1

1

3 1n n

1

31

4 1n

n n

1.

2.

3.

4.

5.

6.

1nenn