Page 196 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
2 - (n 1)x 2
-
-
nx
Example: Let f (x) = n x e – (n – 1) x e , x [0, 1].
n
n
Consider the series å f (x) .
n
=
k 1
n 2 2 2
-
In this case S (x) = å (k x - kx - (k - 1)xe - (k 1)x ) = n x e - nx
n
k 1
=
As you have seen that this sequence (S,) is pointwise but not uniformly convergent to the
function f where f(x) = 0, x (0, 1). Thus the series f (x) is pointwise convergent but not
n
uniformly to the function f where f(x) = 0, x [0, 1].
There is a very useful method to test the uniform convergence of a series of functions. In this
method, we relate the terms of the series with those of a series with constant terms. This method
is popularly called Weierstrass’s M-test given by the German mathematician K.W.T. Weierstrass
(1815-1 897). We state this test in the form of the following theorem (without proof) and illustrate
the method by an example.
Theorem 5: Weierstrass M-Test
Let f be a series of functions defined on a subset A of R and let (M ) be a sequence of real
n n
numbers such that M is convergent and |f ,(x)|  M , " n and " x A. Then f is uniformly
n n n n
and absolutely convergent on A.
¥ x
Example: Test the uniform convergence of the series å
2
n 1 n (n 1)+
=
x k
"
Solution: Since |f (x)|= 2  3 , n and [0, k] .
n
n (n 1) n
+
k
Now the series M = k  is known to be convergent, by p-test.
n 3
n
Therefore, by Weierstrass M-test, the given series is uniformly convergent in the set [0, k].

¥ 1
Task Show that the series å converges uniformly, " x R.
4
n 1 n + x 2
=
Self Assessment
Fill in the blanks:

1. A sequence of functions (f ) defined on a set A is said to be convergent pointwise to f if for
n
each x in A, we have given fn (x) = f(x). Generally, we write ............................. on A.
2. A sequence of functions (f ) defined on a set A is said to be uniformly convergent to a
n
function f on A if given a number  > 0, there exists a positive integer m depending only
on  such that .............................
3. If (f ) be a sequence of continuous functions defined on [a, b] and (f ) ® f uniformly on [a,
n n
b], then f is continuous on [a, b] is known as .............................

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