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Real Analysis




                    Notes                      1           1       1       1
                                   Therefore,  lim f (x) dx =  1    ò  f(x) dx = ò f(x) dx = ò lim (f (x))dx . That is, the integral of the limit
                                                n ò
                                                                                n
                                            n®¥         2                   n®¥
                                               0           0       0       0
                                   is not equal to be limit of the sequence of integrals. In fact, (f ) is not uniformly convergent to f
                                                                                   n
                                   in [0, 1]. This we prove by the contradiction method. If possible, let the sequence be uniformly
                                                              1                                              1
                                   convergent in [0, 1]. Then, for  =   , there exists a positive integer m such that |f(x) – f(x)| <   ,
                                                              4                                              4
                                   for n  m and  " x [0, l].
                                       nx   1
                                   i.e.,   2   <   , for n  m and V x [0, 1]
                                      e nx  4
                                                                      1
                                   Choose a positive integer M  m such that    [0, 1],
                                                                      M

                                                   1
                                   Take n = M and x   . We get
                                                   M

                                                   1    1        e  2
                                                      <    i.e., M <    < 1.
                                                   M    4        16
                                   which is a contradiction. Hence (f) is not uniformly convergent in [0, 1].
                                   Now we give the theorems without proof which relate uniform convergence with continuity,
                                   differentiability and integrability of the limit function of a sequence of functions.
                                   Theorem 2: Uniform Convergence and Continuity

                                   If (f ) be a sequence of continuous functions defined on [a, b] and (f ) ® f uniformly on [a, b], then
                                     n                                                 n
                                   f is continuous on [a, b].
                                   Theorem 3: Uniform Convergence and Differentiation
                                   Let (f ) be a sequence of functions, each differentiable on [a, b] such that (f(x )) converges for some
                                       n                                                    n  0
                                   point x of [a, b]. If (f ) converges uniformly on [a, b] then (f ) converges uniformly on [a, b] to a
                                        0          n                              n
                                   function f such that
                                          f’(x) = lim f’ (x); x [a, b].
                                               n®¥  n
                                   Theorem 4: Uniform Convergence and Integration
                                   If a sequence (f ) converges uniformly to f on [a, b] and each function f  is integrable on [a, b],
                                               n                                            n
                                   then f is integrable on [a, b] and
                                       b           b
                                       J f(x) dx = lim J  f (x) dx
                                                     n
                                       a       n®¥ a
                                   14.3 Series of Functions

                                   Just as we have studied series of real numbers, we can study series formed by a sequence of
                                   functions defined on a given set A. The ideas of pointwise convergence and uniform convergence
                                   of sequence of functions can be extended to series of functions.







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