Page 266 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
n 1 1 1 1
Therefore, lim å =  - dx. because lim = 0.
n¥ r 1 2 0 1 x 2 n¥ n
-
=
æ r ö
n 1 + ç ÷
è n ø
p
The value of this integral, on the r.h.s. of last equality, is ,
2
3 n n 2
Example: Find lim å n 3 .
n¥ r 1 (3 + r)
=
Solution: We have

æ ö
n 2 1 ç 1 ÷
= ç . ÷
n
(3 + r) 3 n ç æ r ö 3 ÷
ç ç 3 + ÷ ÷
è è n ø ø
n
Since the number of terms in the summation is 3 , the resulting definite integral will have the
limits from 0 to 3.
3 n n 2 3 n 1 1 3 dx
Therefore, lim å 3 = lim å  =
n¥ r 1 ( 3 + r ) n¥ r 1 n (3 r)+ 3 0 (3 x)+ 3
=
=
n
This integral you can evaluate easily.
Self Assessment

Fill in the blanks:
1. Let P, and P, be two partitions of [a,b]. We say that P, is finer than P, or P , refines P, or P
2 2
is a refinement of P if ................., that is, every point of P is a point of P,.
1 1
2. Let f: [a,b]  R be a bounded function. The infimum or the greatest lower hound of, the set
of ail upper sums is called the upper (Riemann) integral of f on [a, b] and is denoted
by,..........................................

3. If the partition P is a refinement of the partition P, of [a,b], then L(P ,f) £ L(P ,f) and
2 1 2
.............................................
4. The integrability is not affected even when there are finites number of points of
................................ or the set of points of discontinuity of the function has a finite number
of limit points.
5. If f : [a, bJ  R is a ......................, then f is integrable over [a, b], that is f Î R(a,b).

20.5 Summary

 In this unit, you have been introduced to the concept of integration without bringing in
the idea of differentiation. As upper and lower sums and integrals of a bounded function
f over closed interval [a,b] have been defined. You have seen that upper and lower Riemann
integrals of a bounded function always exist. Only when the upper and lower Riemann
integrals are equal, the function f is said to be Riemann integrable or simply integrable
over [a,b] and we write it as f Î R [a,b] and the value of the integral of f over [a,b] is

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