Page 268 - DMTH401_REAL ANALYSIS

Real Analysis

Notes 4. Prove that the function f defined in [0, 1] by the condition that if r is a positive integer,
1 1
r-1
f(x) = (–1) when < x £ , and f(x) = 0, elsewhere, is integrable.
r 1 r
+
1 1 1
5. Show that the function f defined in [0, 1], for integer a > 2, by f(x) = , when < x <
-
-
a r 1 a r a r 1
1
(r = 1,2,3) , and f(0) = 0, is integrable.
-
a r 1
6. Give example of functions f and g such that f – g, fg, f/g are integrable but f and g may not
be integrable over [a, b].
7. Find the limit, when n tends 10 infinity, of the series
n n n n
+ + +¼+
n 3 (n + 4) 3 (n 8) 3 [n + 4(n 1)] 3
-
+
8. Find the limit, when 11 tends to infinity, of the series
1 1 1 1
+ + +¼+ .
n n 1 n + 2 3n
+
Answers: Self Assessment

h
1. P  P 2.  f(x)dx.
1 2
a
3. U(P ,f) £ U(P ,f) 4. discontinuity
2 1
5. continuous function

20.8 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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