The ratio and root test

(As in the previous example.)

Recall: There are three possibilities for power series convergence.

1 The series converges over some finite interval:(the interval of convergence).

The series may or may not converge at the endpoints of the interval.

There is a positive number R such that the series diverges for but converges for .x a R x a R

2 The series converges for every x. ( )R

3 The series converges for at and diverges everywhere else. ( )0R

x a

The number R is the radius of convergence.

Ratio Technique

We have learned that the partial sum of a geometric series is given by:

111

n

nrS tr

where r = common ratio between terms

When , the series converges.1r

Geometric series have a constant ratio between terms. Other series have ratios that are not constant. If the absolute value of the limit of the ratio between consecutive terms is less than one, then the series will converge.

For , if then:1

nn

t

1lim n

nn

tLt

if the series converges.1L

if the series diverges.1L

if the series may or may not converge.1L

The series converges if .1L

The series diverges if .1L

The test is inconclusive if .1L

The Ratio Test:

If is a series with positive terms andna 1lim n

nn

aL

a

then:

Determine if the series converges

Does the series converge or diverge?

Does the series converge or diverge?

Series diverges

The series converges if .1L

The series diverges if .1L

The test is inconclusive if .1L

Nth Root Test:

If is a series with positive terms andna lim nnn

a L

then:

Note that the rules are the same as for the Ratio Test.

Helpful tip

When using the root test we often run into the limit nth root of n as n approaches ∞ which is 1

(We prove this at the end of the slide show)

example: 2

1 2nn

n

2

2n

n

n 2

2

n n

2lim n

nn

2lim n

nn

?

example: 2

1 2nn

n

2

2n

n

n 2

2

n n

2

lim2

n

n

n

2lim n

nn

2lim n

nn

21 1

12

it converges

?

another example:2

1

2n

n n

2

2nn

n 2

2n n

2

2limnn n

21

it diverges2

Tests we know so far:Try this test firstnth term test (for divergence only)Then try theseSpecial series: Geometric, Alternating, P series, TelescopingGeneral tests: Ratio TestDirect comparison test, Limit comparison test,Root testIntegral test, Absolute convergence test (to be used with another test)

HomeworkP 647 13-31 odd,51-65 odd87-92 all

How can you measure the quality of a bathroom?Use a p-series test

By Mr. Whitehead

lim n

nn

1

lim nn

n

1lim ln nn ne

1lim lnn

nne

lnlimn

nne

1

lim1nn

e

0e1

Indeterminate, so we use L’Hôpital’s Rule

formula #104

formula #103

Extra example of ratio test

Does the series converge or diverge?

Does the series converge or diverge?Extra example of the ratio test