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un x u1 x u2 x u3 x
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is said to be convergent in a; b if the sequence of partial sums fSn x g, n 1; 2; 3; . . . ; where Sn x u1 x u2 x un x , is convergent in a; b . In such case we write lim Sn x S x n!1 and call S x the sum of the series. It follows that un x converges to S x in a; b if for each  > 0 and each x in a; b we can nd N > 0 such that jSn x S x j <  for all n > N. If N depends only on  and not on x, the series is called uniformly convergent in a; b . Since S x Sn x Rn x , the remainder after n terms, we can equivalently say that un x is uniformly convergent in a; b if for each  > 0 we can nd N depending on  but not on x such that jRn x j <  for all n > N and all x in a; b . These de nitions can be modi ed to include other intervals besides a @ x @ b, such as a < x < b, and so on. The domain of convergence (absolute or uniform) of a series is the set of values of x for which the series of functions converges (absolutely or uniformly).
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EXAMPLE 1. Suppose un xn =n and 1 @ x @ 1. Now think of the constant function F x 0 on this interval. 2 For any  > 0 and any x in the interval, there is N such that for all n > Njun F x j < , i.e., jxn =nj < . Since the limit does not depend on x, the sequence is uniformly convergent.
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EXAMPLE 2. If un xn and 0 @ x @ 1, the sequence is not uniformly convergent because (think of the function F x 0, 0 @ x < 1, F 1 1 jxn 0j <  when xn < ; thus n ln x < ln :
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On the interval 0 @ x < 1, and for 0 <  < 1, both members ln  : Since of the inequality are negative, therefore, n > ln x ln  ln 1 ln  ln = , it follows that we must choose N ln x ln 1 nn x ln 1=x such that n>N> ln 1= ln 1=x
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1 From this expression we see that  ! 0 then ln ! 1 and  1 also as x ! 1 from the left ln ! 0 from the right; thus, in either x case, N must increase without bound. This dependency on both  and x demonstrations that the sequence is not uniformly convergent. For a pictorial view of this example, see Fig. 11-1.
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Fig. 11-1
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SPECIAL TESTS FOR UNIFORM CONVERGENCE OF SERIES 1. Weierstrass M test. If sequence of positive constants M1 ; M2 ; M3 ; . . . can be found such that in some interval (a) jun x j @ Mn n 1; 2; 3; . . . (b)
1 X cos nx n 1
Mn converges
then un x is uniformly and absolutely convergent in the interval.
EXAMPLE. converges. n2
  X1 cos nx 1 is uniformly and absolutely convergent in 0; 2 since  2  @ 2 and  n  n n2
This test supplies a su cient but not a necessary condition for uniform convergence, i.e., a series may be uniformly convergent even when the test cannot be made to apply. One may be led because of this test to believe that uniformly convergent series must be absolutely convergent, and conversely. However, the two properties are independent, i.e., a series can be uniformly convergent without being absolutely convergent, and conversely. See Problems 11.30, 11.127. Dirichlet s test. Suppose that (a) (b) the sequence fan g is a monotonic decreasing sequence of positive constants having limit zero, there exists a constant P such that for a @ x @ b ju1 x u2 x un x j < P Then the series a1 u1 x a2 u2 x is uniformly convergent in a @ x @ b.
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