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2
Let be an infinite series of positive terms.
The series converges if and only if the sequence of partial sums,
, converges. This means:
Definition of Convergence for an infinite series:
1nna
aaaaSn 321
1
limn
nnn
aS
If , the series diverges.
Divergence Test:0lim
nna
1nna
Example: The series
12 1n n
nis divergent since 1
11
1lim
1lim
22
nn
nnn
However, 0lim n
na does not imply convergence!
4
Geometric Series:
The Geometric Series: converges for
If the series converges, the sum of the series is:
Example: The series with and converges .
The sum of the series is 35.
12 nararara 11 r
r
a
1
n
n
1 8
75
8
351 aa
8
7r
5
p-Series:
The Series: (called a p-series) converges for
and diverges for
Example: The series is convergent. The series is divergent.
1p
1
1
npn1p
1001.1
1
n n
1
1
n n
8
Integral test:
If f is a continuous, positive, decreasing function on with
then the series converges if and only if the improper integral converges.
Example: Try the series: Note: in general for a series of the form:
13
1
n n
),1[ nanf )(
1nna
1
)( dxxf
9
Comparison test:
If the series and are two series with positive terms, then:
(a)If is convergent and for all n, then converges.
(b)If is divergent and for all n, then diverges.
• (smaller than convergent is convergent)• (bigger than divergent is divergent)
Examples: which is a divergent harmonic series. Since the original series is larger by comparison, it is divergent.
which is a convergent p-series. Since the
original series is smaller by comparison, it is
convergent.
1nna
1nnb
1nnb nn ba
1nna
1nnb nn ba
1nna
22
22
2
13
3
2
3
nnn nn
n
n
n
1
21
31
23
1
2
5
2
5
12
5
nnn nn
n
nn
n
10
Limit Comparison test:
If the the series and are two series with positive terms, and if
where then either both series converge or both series diverge.
Useful trick: To obtain a series for comparison, omit lower order terms in the numerator and the denominator and then simplify.
Examples: For the series compare to which is a convergent p-series.
For the series compare to which is a divergent geometric series.
1nna
1nnb c
b
a
n
n
n
lim
c0
12 3n nn
n
1 2
31
2
1
nn nn
n
123n
n
n
n
n
11 33 n
n
nn
n
11
Alternating Series test:
If the alternating series satisfies:
and then the series converges.
Definition: Absolute convergence means that the series converges without alternating (all signs and terms are positive).
Example: The series is convergent but not absolutely convergent.
Alternating p-series converges for p > 0.
Example: The series and the Alternating Harmonic series are convergent.
1
65432111
nn
n bbbbbbb
1 nn bb 0lim n
nb
0 1
1
n
n
n
1
)1(
np
n
n
1
)1(
n
n
n
1
)1(
n
n
n
12
Ratio test:
(a)If then the series converges;
(b)If the series diverges.
(c)Otherwise, you must use a different test for convergence.
If this limit is 1, the test is inconclusive and a different test is required.
Specifically, the Ratio Test does not work for p-series.
Example:
1lim 1
n
n
n a
a
1nna
1lim 1
n
n
n a
a
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Summary:Apply the following steps when testing for convergence:
1.Does the nth term approach zero as n approaches infinity? If not, the Divergence Test implies the series diverges.
2.Is the series one of the special types - geometric, telescoping, p-series, alternating series?
3.Can the integral test, ratio test, or root test be applied?
4.Can the series be compared in a useful way to one of the special types?
16
Example
Determine whether the series converges or diverges using the Integral Test.
Solution: Integral Test:
Since this improper integral is divergent, the series
(ln n)/n is also divergent by the Integral Test.
19
Example 2: Using the limit comparison test
Since an is an algebraic function of n, we compare the given series with a p-series.
The comparison series for the Limit Comparison Test is
where