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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_13” 2011/5/22 23:35 page 52 #52
13-52 Real Analysis
Since
x
1/3
1 x
log
1
x
= x
1/3
log
1
x
n=0
x
n
(0 < x < 1)
Theorem 13.4.9 gives
1
0
x
1/3
1 x
log
1
x
dx =
n=0
1
0
x
n+1/3
log
1
x
dx =
n=0
9
(3n + 4)
2
.
(Use integration by parts after the change of variable, log x = t).
4. Compute
0
sin t
e
t
x
dt, 1 x 1.
sin t
e
t
x
= lim
N →∞
N
n=0
x
n
e
(n+1)t
sin t.
However for t > 0,
"
"
"
"
"
N
n=0
x
n
e
(n+1)t
sin t
"
"
"
"
"
te
t
1 x
N +1
e
(N +1)t
1 xe
t
2t
e
t
x
= g(x).
Since g(x) is an integrable function, Theorem 13.5.8 applies to
the sequence of partial sums, giving
0
sin t
e
t
x
dt =
n=0
x
n
0
e
(n+1)t
sin tdt=
n=0
x
n
1 + (n + 1)
2
.
(Use integration by parts twice).
13.6 GENERALIZA ...
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Publisher Resources

ISBN: 9781299447561Publisher Website