The University of Sydney
School of Mathematics and Statistics
Tutorial 5 (Week 6)
MATH2962: Real and Complex Analysis (Advanced)
Semester 1, 2016
Web Page: http://www.maths.usyd.edu.au/u/UG/IM/MATH2962/
Lecturer: Florica C. Cı̂rstea
Questions marked with * are more difficult questions.
Material covered
(1) Convergence tests for series
Outcomes
This tutorial helps you to
(1) know and be able to apply convergence tests for sequences and series
Summary of essential material
P
Series are sequences formed by P
summing terms ak ∈ KN . We denote series by ∞
k=0 ak . The
n
sequence (sn ) defined
P by sn := k=0 for all n ∈ N is called the sequence of partial sums We
say that the series ∞
k=0 ak is convergent if the sequence of partial sums (sn ) converges, and it
is divergent if the sequence of partial sums (sn ) diverges.
Convergence and divergence of series is determined by applying convergence tests.
Cauchy criterion
• Applies to: All series
• Useful mostly for theoretical purposes, rarely used on specific series.
P∞
For every ε > 0 there exists n0 ∈ N so that
k=0 ak < ε for all n > n0 .
Comparison and limit comparison tests
• Applies to: Series with non-negative terms.
• Useful for series in R and for testing for absolute convergence (apply to
P∞
k=0
kak ).
• Requires knowledge about elementary series: geometric series, p-series, harmonic
series, exponential series.
Suppose that ak ≥ 0 and bk > 0 for all k ∈ N. Let one of the two conditions be satisfied:
(i) there exists m ∈ N such that ak ≤ bk for all k ≥ m (Comparison test)
ak
(ii)
is a bounded sequence (Limit Comparison test)
bk
P
P∞
P∞
P∞
If ∞
k=0 bk converges, then
k=0 ak converges. If
k=0 ak diverges, then
k=0 bk diverges.
Copyright c 2016 The University of Sydney
1
Leibniz test
• Applies to: alternating series only.
• Useful for alternating series only, does not guarantee absolute convergence.
P
k
Suppose that ak ≥ 0 and ak+1 ≤ ak for all k ∈ N, and that ak → 0. Then ∞
k=0 (−1) ak
converges.
Cauchy condensation test
• Applies to: series with non-negative and decreasing terms.
• Useful for series where the root and ratio test is inconclusive
P∞
Suppose
that
a
≥
0
and
a
≤
a
for
all
k
∈
N.
Then
k
k+1
k
k=0 ak converges if and only if
P∞ k
k=0 2 a2k converges.
Root and ratio tests
• Applies to: series in RN or CN .
• Useful for testing for absolute convergence
Define L by one of the following limits:
kan+1 k
(Ratio test)
n→∞ kan k
p
(ii) L := lim sup n kan k1 (Root test)
(i) L := lim
n→∞
P∞
The series k=0 ak converges absolutely if L < 1, diverges if L > 1, and no conclusion
can be made if L = 1.
P
The table below shows to wich classes of series ∞
k=0 ak the tests are applicable in terms of
properties of ak and the partial sums (sn ).
7
Cauchy criterion
Leibniz test
7
Root test
Integral test
7
Ratio test
Cauchy condensation test
ak ≥ 0 4 4 4
Comparison test
Limit comparison test
7 means that the test is not in general applicable.
boundedness of (sn )
4 means that the test is applicable,
4 4 4
ak ≥ 0 and ak decreasing 4 4 4 4 4
7
4 4 4
ak = (−1)k |ak |, |ak | decreasing
7
7
7
7
7
4 4 4 4
ak ∈ R (arbitrary signs)
7
7
7
7
7
7
4 4 4
ak ∈ RN or CN 7 7 7 7 7
P
4 4 4 4 4
or CN , test on ∞
k=1 kak k
7
4 4 4
7
4 4 4
ak ∈ RN
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Questions to complete during the tutorial
1. Classify the given series as either absolutely convergent, convergent, or divergent.
r
∞
∞
∞
3
X
X
X
n
2n
n
n
nn
(a)
(−1)
;
(c)
(−1)
;
(e)
(−1)
;
2+1
n
n
3n
1
+
4
2
n=0
n=0
n=0
√
∞
∞
∞
X
X
X
n
n!
1
(b)
(−1)n 2
;
(d)
(−1)n n ;
(f)
(−1)n
.
n
3n
+
1
n
(log
n)
n=0
n=1
n=2
2. Use the Cauchy condensation test to determine for which p > 0 the following series
converge.
(a)
∞
X
n=2
1
;
n(log n)p
(b)
∞
X
n=3
∞
X
1
1
; (c)
p
n log n(log(log n))
(log n)p
n=2
P
3. Let (ak ) and (bk ) be sequences in C and let sn := nj=0 aj .
(a) Show that
n
n
X
X
aj bj = bn+1 sn −
sk (bk+1 − bk ).
j=0
(1)
k=0
Hint: Use the definition of sk and write the sum on the right hand side as a double
sum. Then interchange the order of summation.
P
(b) Suppose that (sn ) is bounded, that ∞
k+1 − bk ) converges absolutely, and that
k=0 (bP
bk → 0 as k → ∞. Prove that the series ∞
k=0 aj bj converges. This is known as
Dirichlet’s test.
(c) How does the above Dirichlet test generalise the Leibniz test for alternating series?
(d) Suppose that the sequence of partial sums of the series
that
∞
X
ak
k=1
k
∞
X
ak is bounded. Show
k=1
converges.
Extra questions for further practice
4.
(a) Let a > 0. By using partial fractions, find an expression for the partial sums of
the series
∞
X
1
.
(a
+
k)(a
+
k
+
1)
k=1
Then find its limit.
*(b) Let a > 0. Use partial fractions to write the partial sums of the series
∞
X
k=1
1
(a + k)(a + k + 1)(a + k + 2)
in terms of the partial sums in part (a). Then find its limit.
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5. The purpose of this question is to show that if the terms change sign, the limit comparison
test does not necessarily work.
∞
X
1
(a) Show that the series
(−1)k+1 √ converges.
k
k=1
√
∞
k
X
k+1 k − (−1)
(b) Show that the series
diverges.
(−1)
k
k=1
√
k
1
ak
k+1 k − (−1)
(c) For k ≥ 1, set ak := (−1)
and bk := (−1)k+1 √ . Show that
k
bk
k
has a finite limit as k → ∞, but the limit comparison test does not apply.
Challenge questions (optional)
6. Let A = [aij ] ∈ KN ×N be an N × N matrix. Define a matrix norm by
kAk :=
N X
N
X
|aij |2
1/2
.
i=1 j=1
2
Note that this is simply the usual norm in KN ×N = KN , and hence has all its properties
including the triangle inequality. Prove that the matrix exponential
eA :=
∞
X
1 k
A
k!
k=0
converges absolutely. Hint: Use that kAk k ≤ kAkk by Tutorial 1, Question 9.
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