Page 228 - DMTH401_REAL ANALYSIS

Real Analysis

Notes ¢ ¥ ¢
Differentiability for Series: If (a) all g are differentiable, (b) all g are continuous, (c) å n= 1 g n
n
n
c
b
converges uniformly and (d) c$ Î [ , ] s.t. å ¥ g ( ) < ¥ then å ¥ g converges uniformly
a
n= 1 n n= 1 n
æ ¥ ö ¥
and çå gn ¢ = å g ¢ n .
÷
è n= 1 ø n= 1
17.8 Review Questions
1
2
1. Prove that f :  ®  , f ( ) = sin(n x .
)
x
n n
n
¥ n
2. x on [–1/2, 1/2] and on (–1, 1). We want to investigate convergence
Consider a series å n
= 0
of this series, whether it is pointwise of n uniform and eventually, what is the limit.
¥ sin(nx )
3. Consider series å on . By Weierstrass M-test, we see that this series converges
i= 1 2 n
uniformly on  since
sin(nx ) 1 ¥ 1
£ and å < ¥
2 n 2 n 1 2 n
n=
¥ 1
).
4. Consider series å on [0, ¥ By Weierstrass M-test, we can obtain uniform
2
n= 1 n + x
)
convergence of this series on [0, ¥ since
¥
1 £ 1 and å 1 < ¥
2
n + x n 2 n= 1 n 2
Answers: Self Assessment
1. series of functions 2. { } ¥ 1
f
n n=
3. converges uniformly 4. radius of convergence
5. differentiable

17.9 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).

G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.
Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.
S.C. Malik: Mathematical Analysis.

H.L. Royden: Real Analysis, Ch. 3, 4.

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