Page 198 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 2. Test the uniform convergence of the following sequence of functions in the specified
domains
1
(i) f (x) = in 0 < x < ¥
n nx
nx
(ii) f (x) = , – ¥< < x < ¥
2
n 1 n x 2
+
x n
(iii) f (x) = , 0 x  1
n 1 x n
+
1
(iv) f (x) = , 0  x < ¥
n n
x
3. Show that the limit function of the sequence (f,) where (f ) (x) = , x  R, is continuous in
n n
R while (f,,) is not uniformly convergent.
2 n
4. Show that for the sequence (f,) where (f ) (x) = nx (1 – x ) , x [0, 1], the integral of the limit
n
is not equal to the limit of the sequence of integrals.
5. Show that the series
x x x
+ + +...... is uniformly convergent in ]k, ¥[ where k is a
x 1 (x 1)(2x 1) (2x 1)(3x 1)
+
+
+
+
+
positive number.
x
6. Show that the series X is uniformly convergent in [0, k] where k is any positive
n(n 1)
+
number but it does not converge uniformly in [0, ¥].
Answers: Self Assessment
1. f ® f (pointwise) 2. f (x) – f(x)| <  for n  m and " x A.
n n
3. Convergence and Differentiation 4. f’(x) = lim f’ (x); x [a, b].
n®¥ n
b b
5. J f(x) dx = lim J f (x) dx
n
a n®¥ a
14.7 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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