The ratio and root test
Recall: There are three possibilities for power series convergence.
1
The series converges over some finite interval:
(the interval of convergence).
There is a positive number R such that the series
diverges for x  a  R but converges for x  a  R .
The series may or may not converge at the endpoints
of the interval.
2
The series converges for every x. ( R   )
3
The series converges for at x  a and diverges
everywhere else. ( R  0 ) (As in the previous example.)
The number R is the radius of convergence.

Ratio Technique
We have learned that the partial sum of a geometric series
is given by:
1 r
Sn  t1 
1 r
n
When
where r = common ratio between terms
r  1 , the series converges.

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
t
n 1
n
tn 1
, if L  lim
n  t
n
then:
if
L 1
the series converges.
if
L 1
the series diverges.
if
L 1
the series may or may not converge.

The Ratio Test:
an 1
L
If  an is a series with positive terms and lim
n  a
n
then:
The series converges if
The series diverges if
L 1 .
L 1.
The test is inconclusive if
L  1.

Determine if the series converges
Does the series converge or diverge?
Does the series converge or diverge?
Series diverges
Nth Root Test:
If
a
n
is a series with positive terms and lim n an  L
n 
then:
The series converges if
The series diverges if
L 1 .
L 1.
The test is inconclusive if
L  1.
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
n

n2

n
n 1 2
n
n
n

n
2
2
2

lim n  lim n
n
n 
2
n
n

2
?


n2

n
n 1 2
example:
2
n
n
n
n

n
2
2
2
n
lim
n  2
1

2
2
n 
1
2
n

lim n  lim n
n
2

1
n
n

2
?
it converges


n
2

2
n
n 1
another example:
2  2
n 2
2
n
n
n
n
lim
n  n
2
n
2
2

2
1

it diverges

Tests we know so far:
Try this test first
nth term test (for divergence only)
Then try these
Special series:
Geometric, Alternating, P series, Telescoping
General tests:
Ratio Test
Direct comparison test,
Limit comparison test,
Root test
Integral test,
Absolute convergence test (to be used with another test)
Homework
P 647 13-31 odd,
51-65 odd
87-92 all
How can you measure the quality of a bathroom?
Use a p-series test
By Mr. Whitehead
n
lim n
n 
lim n
formula #104
1
n
n 
e
lim ln
n
e
e
1
lim n
n 1
0
e
1
 
1
nn
1
lim ln n
n n
e
formula #103
ln n
lim
n n
Indeterminate, so we use L’Hôpital’s Rule

Extra example of ratio test
Does the series converge or diverge?
Extra example of the ratio test
Does the series converge or diverge?