Page 195 - DMTH401_REAL ANALYSIS

Unit 14: Sequences and Series of Functions

Definition 4: Series of Functions Notes
A series of the forth f + f + f +......+ f +..... where the f are real functions defined on a given set
1 2 3 n n
¥
ACR is called a series of functions and is denoted by å f . The function f is called nth term of
n
n
n 1
=
the series.
n
For each x in A, f (x) + f (x) + f (x) +....... + is a series of real numbers. We put S (x) = å f (x) . Then
1 2 3 n k
k 1
=
we get a sequence (S,) of real functions defined on A. We say that the given series f + f +....+ f
1 1 n
+..... of functions converges to a function pointwise if the sequence (S„) associated to the given
series of functions converges pointwise to the function f. i.e. (S,, (x)) converges to f(x) for every
x in A.
We also say that f is the pointwise sum of the series f on A.
n
If the sequence (S ) of functions converges uniformly to the function f, then we say that the given
n
series f + f +....... + f +........, of functions converges uniformly to the function f on A and f is
1 2 n
¥
called uniform sum of å f on A. The function S is called the sum of n terms of the given series
n
=
i 1 n
or the n partial sum of the series and the sequence (S ) is called the sequence of partial sums of the
n
¥
series å f . To make the ideas clear, we consider some examples.
n
i 1
=
n-1
Example: Let f (x) = x where x = 1 and –r  x  r where 0 < r < 1. Then the associated
n 0
2
series is 1 + x + x +.....
1 x n
-
2
n–1
In this case, S (x) = 1, + x + x + ......+ x . It is clear that S (x) = .
n n
1 x
-
1
This sequence (S, (x)) of functions is easily seen to converge pointwise to the function f(x) = ,
-
I x
since x ® 0 as n ®¥, since |x| < r < I but the convergence is not uniform as shown below:
n
Let  > 0 be given.
|x| n r n
n
|S (x) – f(x)| =  if r <  (1 – r)
n
-
-
|1 x| 1 r
log ( (1 r))
-

i.e. n >
logr
é log ( (1 r))ù
 -
If m = ê ú = 1, then
ë logr û
|s,(x) – f(x)| <  if n  m and for –r  x  r.
Therefore (S ) converges uniformly in [–r, r]. Thus the geometric series 1 + x + x +..... converges
2
n
I
uniformly in [–r, r] to the sum function f(x) = .
-
1 x
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