Advanced Calculus fi..

41 8 Advanced Calculus, Fifth Edition2. Prove that each of the following series is uniformly convergent over the set of values of xgiven:n.(tanh x)"b)CZ ! + allxsin nxa) C2=l 5, -1 5 x r 1C) C ~ ;;r+ll I d x d) x~log$L) c=~$, -1rxrl.g) CE, nxn, -0.9 5 x 10.9 h) CElnxn, -a 5 x 5 a, a c 11 1f) CElnxn, -2 I X L 2 #I3. Prove: If xzl un(x) is uniformly convergent for a I: x 5 b, then the series is uniformlyconvergent in each smaller interval contained in the interval a 5 x 5 b. More generally,if a series is uniformly convergent for a given set E of values of x, then it is uniformlyconvergent for any set El that is part of E.4. Prove: If Czl vn(x) is uniformly convergent for a set E of values of x and I u ~ ( x ) I 5 vn(x)for x in E, then CF=, un(x) is uniformly convergent for x in E.5. Prove: If 0 < un(x) < lln and u,+~(x) i un(x) for a 5 x 5 b, then the seriesxz,(- 1)"un(x) is uniformly convergent for a 5 x 5 b.6. Prove: If the series Czl M,, of constants Mn is convergent and I fn+,(x) - fn(x)l 5 M,,for x in E, then the sequence fn(x) is uniformly convergent for x in E.7. Prove that the following sequences are uniformly convergent for the range of x given (cf.Problem 6):a) %, 05x51 b) $, -15x51C) '~g('~+"~), 1 5 x 5 2 d) -p, n 21~ x 1 1(B.6.14 PROPERTIES OF UNIFORMLY CONVERGENT SERIESAND SEQUENCESLet C un(x) be a series of functions, each of which is defined for a 5 x 5 b. Let itbe assumed further that this series converges to a sum f (x) for a 5 x 5 b, so thatone hasOne can then ask questions such as the following: If each function un(x) is contin- .uous, is the sum f (x) continuous? If each un(x) has a derivative, does f (x) have aderivative? The following theorems answer such questions.THEOREM 31 The sum of a uniformly convergent series of continuous functionsis continuous; that is, if each un(x) is continuous for a p x 5 b, then so is f (x) =Czl u,(x), provided that the series converges uniformly for a 5 x 5 b.Proof.thatLet xo be given, a 5 xo 5 b, and let 6 > 0 be given. We then seek a S such1 f (x) - f (xo)l < E when Ix - xol < 6and x is in the given interval. We choose N so large that1I - f < 5 , a 5 x 4 b, n t N. (6.28)