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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