Convergence of geometric sequences and series
The sequence ak = ar k converges if and only if −1 < r ≤ 1.


if − 1 < r < 1,
0
k
lim ar = a
if r = 1,

k→∞

DNE if r ≤ −1 or r > 1.
I
Discussion why.
The series
P∞
k=0 ak
∞
X
=
P∞
k=0 ar
k
converges if and only if −1 < r < 1.
ar k = a + ar + ar 2 + · · · =
k=0
a
1−r
if |r | < 1.
The series diverges for |r | ≥ 1.
I
Discussion why.
Math 105 (Section 204)
Series
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Tests for convergence or divergence
While in certain cases it is possible to evaluate a series by coming up with
a formula for its partial sums, e.g.,
∞
X
k=1
∞
X
1
,
k(k + 1)
ar k ,
Sn = 1 −
1
, or
n+1
Sn = a(1 − r n+1 )/(1 − r ),
k=0
in most instances such closed-form expressions are not available. However,
even when one cannot evaluate the series, it is often possible to determine
whether the series converges or diverges, by means of certain tests.
Math 105 (Section 204)
Series
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Divergence test
Statement
If lim ak 6= 0, then
k→∞
∞
X
ak diverges .
k=1
Remarks:
Note that the divergence test is a one-way implication. It does not
say that
∞
X
If lim ak = 0, then
ak converges .
k→∞
k=1
The above statement is false, as one can see from the harmonic
series. (Discussion)
Thus the divergence test can only be used to determine whether a
given infinite series diverges. It cannot be used to establish
convergence.
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Series
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Some examples
Can one use the divergence test on the series
∞
X
k=1
k3
k3 + 1
to conclude that it diverges?
Math 105 (Section 204)
Series
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Some examples
Can one use the divergence test on the series
∞
X
k=1
k3
k3 + 1
to conclude that it diverges?
I
Yes.
How about
∞
X
k2
k=1
Math 105 (Section 204)
Series
ek
?
2011W T2
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Some examples
Can one use the divergence test on the series
∞
X
k=1
k3
k3 + 1
to conclude that it diverges?
I
Yes.
How about
∞
X
k2
k=1
I
ek
?
The test is inconclusive.
Math 105 (Section 204)
Series
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The integral test
Statement
Suppose ak = f (k) for some continuous, positive, decreasing function f .
Then
Z ∞
∞
X
ak and
f (x) dx
1
k=1
either both converge or both diverge.
Remarks:
1. Justification via diagrams
2. Example of the p-series:
∞
X
k −p converges if and only if p > 1.
k=1
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Exercise
For which values of p does the following series converge?
∞
X
k=10
1
k ln k [ln(ln k)]p
A. p > 1
B. p > 0
C. p ≤ 0
D. p < −1
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