Page 264 - DMTH401_REAL ANALYSIS

Real Analysis

Notes x + x x + x x + x
Then, S(P,f) = (x - x ) 1 0 + (x - x ) 2 1 +¼+ (x - x n 1 ) n n 1
-
1
n
-
1
0
2
2 2 2
1
)
2
x -
= é ë ( 1 2 x 0 2 ) (x+ 2 2 - x 1 2 ) +¼+ (x - x 2 n 1 û ù
n
-
2
1 1
2
2
= (x - x 2 0 ) = (b - b 2 ).
n
2 2
1
2
Here again, S(P,f) = (b - a 2 ), no matter what the partition P we may take, Hence
2
b b 1
2
2
 f(x) dx =  x dx = lim S(P,f) = (b - a ).
P 
0
a a 2
The converse of Theorem 15 is also true which we state without proof as the next theorem.
Theorem 16: If a function f is Riemann integrable on a closed interval [a,b], then
b
lim S(P,f) exists and lim S(P,f)=  f(x) dx.
P  0 P  0
a
One of the important application of Theorem 16 is in computing the sum of certain power series.
For, let us consider a partition P of [a,b] having n sub-intervals, each of length h so that nh =
b – a. Then P can be written as P = (a, a + h, a + 2h, ..., a + nh = b).
Let t, = a + ih, i = 1,2 ,...., n. Then
n
+
+
+
S(P,f) å f(t )  x = h[f(a h) f(a 2h) +¼+ f(a nh)].
+
i i
i 1
=
When lim S(P,f) exists, then
P  0
h
+
+
lim h[f(a h) f(a 2h) +¼+ f(a nh)] =  f(x) dx.
+
+
n¥
h 0 a
In the above formulae, we can change the limits of integration from a, b to 0, a, where a Î N. For,
b a
-
by changing h to , it is easy to deduce from above formula that
an
(b a) 1 n é (b a) r ù b
-
-
lim å f a + ú  f(x) dx. (14)
=
a n¥ n = ë ê a hû a
b (b a) a é (b a) ù
-
-
But,  f(x) dx =  f a + x dx.
ê
ú
a a 0 ë a û
Therefore, from (14), we get
1 n é (b a) r ù a é (b a) ù
-
-
lim å f a + ú  f a + x dx. (15)
=
n¥ n r 1 ë ê a n û 0 ê ë a ú û
=
In (15), put a = 0, b = a. We get the following result:
If f is integrable in [0,a], then
n 1 æ ö a
r
lim å f ç ÷  f(x) dx.
=
n
n¥ r 1 n è ø 0
=
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