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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_03” 2011/5/22 22:50 page 42 #42
3-42 Real Analysis
UNSOLVED EXERCISES
1. Let {a
n
}be a sequence of positive real numbers and b
n
=
n
i=1
a
i
. Assume
that
i=1
a
i
=∞.If{x
n
} is a sequence of real numbers such that x
n
x
as n →∞in R, then show that
lim
n→∞
1
b
n
n
i=1
a
i
x
i
= x.
2. Assume that a sequence of positive numbers {b
n
} satisfies 0 < b
1
<
b
2
< b
3
< b
4
< ··· and b
n
→∞as n →∞. If a series
n=1
x
n
of real
numbers converges in R, then show that
lim
n→∞
1
b
n
n
i=1
b
i
x
i
= 0.
In particular, show that if {y
n
} is a sequence of real numbers such
that the series
n=1
y
n
/n converges in R, then (y
1
+ y
2
+ ...y
n
)/n 0
as n →∞.
3. Prove that 2
n
/n!→0asn →∞.
4. Define x
n
=
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Publisher Resources

ISBN: 9781299447561Publisher Website