Math 414: Analysis I
Homework 7
Due: March 28, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: (See Exam 2 Topics for Practice Problems.) Material for Exam 3 will start
with Homework 8.
Turn in the following problems.
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1. Let {an } be a sequence of realPnumbers. Prove that ∞
n=1 (an − an+1 ) converges if
∞
and only if {an } converges. If n=1 (an − an+1 ) converges, what is its sum? (Please
give an explicit answer.)
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P∞ 2
2. Prove that if ∞
n=1 an converges absolutely, then
n=1 an converges absolutely. Is
the converse true? Justify your answer.
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3. (a) Let ∞
n=1 a(n) be such that (a(n)) is a decreasing sequence of strictly positive
numbers. If s(n) denotes the nth partial sum, show (by grouping the terms in
s(2n ) in two different ways) that
1
(a(1) + 2a(2) + . . . + 2n a(2n )) ≤ s(2n ) ≤ a(1) + 2a(2) + . . . + 2n−1 a(2n−1 ) +a(2n )
2
P∞ n n
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Use these inequalities to show that ∞
n=1 2 a(2 )
n=1 a(n) converges if and only if
converges. This result is often called the Cauchy Condensation Test. It is
extraordinarily useful in testing the convergence of series.
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1
(b) Use the Cauchy Condensation Test to discuss the p-series ∞
n=1 np for p > 0.
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1
(c) Use the Cauchy Condensation Test to establish the divergence of ∞
n=2 n ln n .
4. Exercise 2.5.7 in Basic Analysis, Lebl
5. Exercise 2.5.8 in Basic Analysis, Lebl
6. Exercise 2.5.9(b,c) in Basic Analysis, Lebl
7. Exercise 2.5.11 in Basic Analysis, Lebl
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