Section 8.3: The Integral and Comparison Tests; Estimating Sums

Practice HW from Stewart Textbook (not to hand in)

p. 585 # 3, 6-12, 13-25 odd

In this section, we want to determine other methods for determining whether a series converges or diverges.

The Integral TestFor a function f, if f (x) > 0, is continuous and decreasing for and , then

either both converge or both diverge.

Mx )(nfan

MMnn dxxfa )( and

Note

The integral test is only a test for convergence or divergence. In the case of convergence, it does not find a value for the sum of the series.

Example 1: Determine the convergence or divergence of the series

Solution:

1 141

n n

Example 2: Determine the convergence or divergence of the series

Solution: (In typewritten notes)

22)(ln

1 n nn

Example 3: Show why the integral test cannot be used to analyze the convergence or divergence of the series

Solution:

1

1

)1(

n

n

n

p-Series and Harmonic SeriesA p-series series is given by

If p = 1, then

is called a harmonic series.

1 41

31

21

111

npppppn

1 41

31

2111

n n

Convergence of p – series

A p-series

1. Converges if p > 1. 2. Diverges if .

1 41

31

21

111

npppppn

1p

Example 4: Determine whether the p-series

is convergent or divergent.

Solution:

251

161

91

411

Example 5: Determine whether the p-series is convergent or divergent.

Solution:

1

1

n n

Example 6: Determine whether the p-series is convergent or divergent.

Solution:

1

1

n n

Making Comparisons between Series that are Similar

Many times we can determine the convergence or divergence of a series by comparing it with the known convergence or divergence of a related series. For example,

is close to the p-series ,

is close to the geometric series .

12 231

n n

12

1

n n

1 134

nn

n

1 34

nn

n

Under the proper conditions, we can use a series where it is easy to determine the convergence or divergence and use it to determine convergence or divergence of a similar series using types of comparison tests. We will examine two of these tests – the direct comparison test and the limit comparison test.

Direct Comparison Test Suppose that and are series only with positive terms ( and ).

1. If is convergent and for all terms n, is convergent.

2. If is divergent and for all terms n, is divergent.

na

na

na nb

nb

nb

0na 0nb

nn ba

nn ba

Example 7: Determine whether the series is convergent or divergent.

Solution:

12 231

n n

Example 8: Determine whether the series is convergent or divergent.

Solution:

1 134

nn

n

Example 9: Demonstrate why the direct comparison test cannot be used to analyze the convergence or divergence of the series

Solution:

02 11)1(

n

n

n

Limit Comparison Test

Suppose that and are series only with positive terms ( and ) and

where L is a finite number and L > 0.

Then either and either both converge or and both diverge.

na nb

0na 0nb

Lba

n

nn

lim

na nb

nb na

Note: This test is useful when comparing with a p-series. To get the p-series to compare with take the highest power of the numerator and simplify.

Example 10: Determine whether the series is convergent or divergent.

Solution:

12 52

35

n nnn