(2) n1 ln(1 n)
Sol. (1) absolutely convergent (2) conditionally convergent
The ratio test
The ratio test
(1) If
lim an1 a n
n
L 1,
then an
n 1
is absolutely convergent.
converges or diverges.
n 1
(a 0)
Sol.
an
ln 1
a n
ln 1 ln a
e n
1 nln a
diverge for 0 a e
converge for a e
Alternating series
An alternating series is a series whose terms are
converge or both diverge.
(ii) when c=0, then the convergence of bn implies the convergence of an. (iii) when c , then the divergence of bn implies the
(ii) If bn is divergent and an bn for all n, then an is also
divergent.
1
Ex. Determine whether n1 2n 1 converges.
11 Sol. 2n 1 2n
So the series converges.
The comparison tests
Theorem Suppose that an and bn are series with positive terms, then
(i) If bn is convergent and an bn for all n, then an is
also convergent.
where bn is a positive number.
The alternating series test
Theorem If the alternating series
(1)n1bn b1 b2 b3 b4 b5 b6 L
n1
satisfies (i) bn1 bn for all n
then
lim
n
an
/ bn
2
(2) diverge. take bn 1/ n
then
lim
n
an
/ bn
(3) converge for p>1 and diverge for p 1 take bn 1/ n p
then
lim
n
an
/ bn

p
Question
ln 1
Ex. Determine whether the series a n
(1)
n1
(1)n1 n
( 0)
(2)
n1
(1)n1 n2 n3 1
Sol. (1) converge (2) converge
Question. (1)n1n
n1 4n 1
Absolute convergence
A series an is called absolutely convergent if the series of absolute values | an | is convergent.
Theorem. If a series is absolutely convergent, then it is convergent.
Example
Ex. Determine whether the following series is convergent.
sin n
(1)
n1
n2
(1)n
(1)n1
Fwohrileexathmepallet,erthneatsinergiehsanrm1 onni3c/
2 is series
absolutely is not.
convergent
A series an is called conditionally convergent if it is convergent but not absolutely convergent.
divergence of an.
Example
Ex. Determine whether the following series converges.
(1)
2n2 3n
n1 5 n5
1
(2) n1 ln2 (n 1)
(3) sin p
n1
n
Sol. (1) diverge. choose bn 1/ n1/2
(ii)
lim
n
bn
0
Then the alternating series is convergent.
(bn 0)
Ex. The alternating harmonic series (1)n1
is convergent.
n1 n
Example
Ex. Determine whether the following series converges.
(2) If
lim an1 a n
n
L 1 or
lim an1 a n
n
then an diverges.
n 1
(3) If lim an1 1, the ratio test is inconclusive: that is, no a n
alternatively positive and negative. For example,
1 1 1 1 L (1)n1
234
n1 n
The n-th term of an alternating series is of the form
an (1)n1bn or an (1)n bn
The limit comparison test
Theorem Suppose that an and bn are series with
positive terms. Suppose
lim an c.
Then
b n n
(i) when c is a finite number and c>0, then either both series