1432
Dont Math forget online quizzes.
THINK: Find the number to replace the question mark.
369542 is to 246359 as
172896 is to 268179 as
417638 is to ?
Review:
Geometric Series Test.
The geometric series diverges if r 1 .
a1
If r < 1 , then the series converges to the sum S =
.
1r
The sum of the first n terms of a geometric series is S =
Ex.
n =1
3
( 1 .2 )
n
(
a1 1 r n
1r
).
Basic Divergence Test
If l i m a n 0 , then the series
n
Ex.
a
n =1
n2 1
n
n =1
2
+n
n
diverges.
p-Series Test:
A series of the form
1
n
n =1
p
=
1
1
p
+
1
2
p
+
1
3
p
+ ... +
The p-series diverges if 0 < p 1.
The p-series converges if p > 1.
Ex.
n =1
27
n
8
n =1
1
n
2
3
1
n
p
+ ...
Integral Test:
If f is positive, continuous, and decreasing for x 1 and a n = f ( n ) ,
then
a
n
and
n =1
Ex.
n=2
f ( x ) dx either both converge or both diverge.
1
n ln( n )
1
Basic Comparison Test:
If a n 0 and b n 0 and
1) If
b
n
converges and 0 a n b n , then
n =1
a
n =1
Ex.
n
n =1
1
2
+n
a
n
converges.
n =1
2) If
n
diverges and 0 a n b n , then
b
n =1
n
diverges.
Limit Comparison Test:
Let
a
n
and
an
and li m
n
bn
b
n
be series with positive terms ( a n > 0 , b n > 0 )
= L , where L is both finite and positive.
Then the two series
a
n
and
b
n
either
both converge or both diverge.
Ex.
n
n =1
n
3
+1
Section 11.3
The Root Test; The Ratio Test
Root Test
Let
a
n
be a series with nonzero terms.
n=b
1.
a
n
converges if l i m n
n
diverges if l i m n
n
n=b
an
2.
a
n=b
n
an
< 1.
> 1.
3. The Root Test in inconclusive if l i m n
n
an
= 1.
(Use another test.)
The Root Test works well for series involving an nth power.
Examples:
1.
n =1
e2n
nn
Determine whether the following series converge or
diverge.
2.
3n + 4 2n
n =1
n
3.
n3
3
n =1
n
Ratio Test Let
a
n
be a series with nonzero terms.
n=b
1.
a
n
converges if l i m
n
n=b
2.
a
n=b
n
diverges if l i m
n
a n+1
an
a n+1
an
< 1.
> 1.
3. The Ratio Test is inconclusive if l i m
n
a n+1
an
=1
(Use another test.)
Series involving factorials and exponential functions work especially
well in the Ratio Test.
Examples:
1.
n=0
2n
n!
Determine whether the following series converge or
diverge.
2.
n=0
n 2 2 n+1
3n
3.
n=0
( n + 1 )!
3n
Popper
1. Determine whether the following sequence converges or diverges.
If it converges, find its limit.
n
5
an = 1
n
a. Diverges
b. Converges, L = e5
c. Converges, L = e1/5
d. Converges, L = e5
2.
(
l n n2 + 4
lim
n
n2 + 3
a.
b.
c.
d.
e.
3/4
1
0
4/3
) =
3.
1
n
n =1
5
a. converges
b. diverges
4.
n =1
1
n
a. converges
b. diverges
5.
5
k =1
1
k 1
a. converges
b. diverges
6.
3n 2 1
1 0n + 5n
n=2
a. converges
b. diverges
2
7.
n
n =1
n
2
+1
a. converges
b. diverges
8.
3
2
n =1
n
a. converges
b. diverges
9.
n
n =1
1
2
+ 4n + 3
a. converges
b. diverges
10.
n=2
1
n2 1
a. converges
b. diverges
11.
n =1
e2n
n
a. converges
b. diverges
12.
2n
2
n + 6n n =1
2
a. converges
b. diverges
13.
n =1
( 1 )
n
3n
a. converges
b. diverges
14.
n
ln( n )
n =1
a. converges
b. diverges
15.
1
1
n
n =1
a. converges
b. diverges
n
16. Choose the series that converges:
a)
n =1
n
2n + 3
3
b)
5
2
n=0
c)
n =1
d)
n =1
2 n+ 3
5 n+1
n
n
n
17.
1
n
n =1
a. converges
b. diverges
18.
n =1
1
n
a. converges
b. diverges
19.
( 1 )
n
n =1
a. converges
b. diverges
2
1
n
20.
n =1
5n
3n + 4 n
a. converges
b. diverges
n =1
21.
si n ( 1 / n )
n
a. converges
b. diverges