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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_09” 2011/5/22 23:21 page3—#3
Sequences and Series of Functions 9-3
3. Let f
n
(x) = (sin nx)/n (x R). The above sequence is defined
for all x
R and lim
n→∞
f
n
(x) = 0 for all x R. In addition,
|f
n
(x)|≤
1
n
0asn →∞for all x R. Thus f
n
(x) f (x)
uniformly on
R as n →∞.
4. Let f
n
(x) = nx for x > 0. This sequence does not converge even
pointwise for any x > 0.
5. The series
n=0
x
2
/(1 + x
2
)
n
converges to
f (x) =
0 for x = 0
1 + x
2
for x = 0
as can be easily verified. This series, however, does not converge
uniformly and this will be proved later.
6. The series
n=0
1
2
n
x
2
1 + x
2
n
(x R)
converges uniformly to
1 +
x
2
2 + x
2
This will also be proved ...
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Publisher Resources

ISBN: 9781299447561Publisher Website