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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_10” 2011/5/22 23:26 page 31 #31
Power Series and Special Functions 10-31
We shall now compute
1
0
x
n
(log x)
n
dx using integration by parts n-
times. Indeed,
1
0
x
n
(log x)
n
dx =
1
0
(log x)
n
d
x
n+1
n + 1
=−
n
n + 1
1
0
x
n
(log x)
n1
dx
=
n(n 1)
(n + 1)
2
1
0
x
n
(log x)
n2
dx
=···=
(1)
n
(n + 1)
n+1
n!
(note that the function x log x is bounded near x = 0 and hence the
integral is the ordinary Riemann integral). We now have
1
0
x
x
dx = 1 +
n=1
1
(n + 1)
n+1
=
n=1
n
n
.
UNSOLVED EXERCISES
1. Let R
1
and R
2
be the radii of convergence of
n=0
a
n
x
n
,
n=0
b
n
x
n
,
respectively and b
n
= 0 for n = 0, 1, 2, .... Show that
(a) if R
1
, R
2
(0, ), then the radius of convergence
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Publisher Resources

ISBN: 9781299447561Publisher Website