Warm Up

Tests for Convergence:

The Integral and P-series Tests

The Integral Test

If f is positive, continuous, and decreasing for x > 1 and an = f(n), then

and

Either both converge or diverge.

1n

n

a

1

( )f x dx

Ex 1:2

1 1n

nn

Ex 2:2

1

11n n

The p-series test

The p-series

Converges if p >1Diverges if 0 < p < 1Test cannot be used if p < 0

1

1p

n n

Ex 3:1

1n n

This is known as the HARMONIC series.

The harmonic series DIVERGES.

Ex 4:5/3

1

2n n

Ex 5:3/ 2

1

4n n

Ex 7: 1 1 11 ...2 2 3 3 4 4

Ex 6:2

1lnn n n

Mixed PracticeMixed PracticeDetermine whether each series converges or diverges. If it converges and is possible to tell, determine what number it

converges to.

1

4( 2)n n n

1

223

n

n

1

23 5n n

2

1

51n

nn

1.

2.

3.

4.

5.

6.1 1 11 ....4 9 16 2

1

n

n

ne