WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (SEQUENCES OF FUNCTIONS, SERIES OF FUNCTIONS & POWER SERIES)

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1 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (SEQUENCES OF FUNCTIONS, SERIES OF FUNCTIONS & POWER SERIES) INSTRUCTOR: CEZAR LUPU Problem. Decide which of the following sequences of functions converge uniformly and which not: (a) f n : [, ] R, f n () = + n 2, n. 2 (b) f n : [0, ] R, f n () = n ( n ), n. (c) f n : [0, ) R, f n () = + n, n. (d) f n : [a, b] R, f n () = + n, n. (e) f n : (, ) R, f n () = n, n. (f) f n : [, ] R, f n () = e 2 /n, n. (g) f n : [0, ) R, f n () = n e /n, n. (h) f n : [0, ] R, f n () = n( ) n, n. University of Pittsburgh Preliminary Eam, 2007 Problem 2. [Dini] Let (f n ) n be a monotonic sequence of continuous functions f n : [a, b] R R such that f n converges pointwise at f. Show that f n converges uniformly. University of Pittsburgh Preliminary Eamination, 2007 Problem 3. Let (g n ) n be a sequence of twice differentiable functions on [0, ] such that for all n, g n (0) = 0 and g n(0) = 0. Suppose that g n() for all n,. Prove that there is a subsequence of (g n ) which converges uniformly. Problem 4. (a) Show that the sequence of functions f n : [, 2] R, defined by (log )n f n () = + (log ) n converges uniformly. (b) Show that the sequence of functions f n : [0, ] R, defined by ( + )n f n () = e 2n converges uniformly on any interval [0, a], with 0 < a <.

2 2 INSTRUCTOR: CEZAR LUPU Problem 5. Consider the sequence of functions f n : [0, ] R which is given by the recursion: f 0, f n+ = f n () + 2 [ f 2 n()], [0, ], n. Show that (f n ) converges uniformly to f() =. Ohio State University Qualifying Eam, 203 Problem 6. Let f : [ a, a] R be a continuous function such that f(0) = 0 and f() <, 0. Consider the sequence of functions (f n ) n, f n : [ a, a] R defined by f = f and f n+ = f f n = f n f for all n. Show that f n converges uniformly to f = 0 on [ a, a]. Problem 7. Define the function ζ by ζ() = Prove that ζ() is defined and has continuous derivatives of all orders in the interval < <. Problem 8. Prove that f() = n. ( ) n+ log Berkeley Preliminary Eam, 985 ( + ) n is defined and differentiable on the open interval < <. Ohio State University Qualifying Eam, 2005 Problem 9. Find all continuous functions f : R R such that there eists a sequence of polynomials P : R R such that P n converges uniformly to f. Problem 0. For each positive integer n, define f n : R R by f n () = cos n. Prove that the sequence of functions f n has no uniformly convergent subsequence. Berkeley Preliminary Eam, 995 Problem. Let the functions f n : [0, ] [0, ], n =, 2,... satisfy f n () f n (y) y whenever y n. Prove that the sequence (f n) has a uniformly convergent subsequence. Berkeley Preliminary Eam, 200

3 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (SEQUENCES OF FUNCTIONS, SERIES α Problem 2. Prove that is uniformly convergent on [0, ) if n(n2 + 3 ) α = 2, but no uniformly convergent on [0, ) if α = 3. University of Pittsburgh Preliminary Eam, 203 Problem 3. (, ): Determine whether the given series is uniformly convergent on (a) (b) (c) (d) (e) 2 n 2 (n + 2 ). n(n + 2 ). 2 n(n ). 2 n(n ). 2 n 2 (n + 3 ). Problem 4. Let f n : R R be differentiable for each n =, 2,... and such that f n() for all n,. Assume lim f n () = g() for all. Show that g is n continuous. Problem 5. Prove directly that if a sequence of continuous functions f n : [0, ] R is equicontinuous and convergent at every point [0, ], then f n is uniformly convergent on [0, ]. Problem 6. Prove the following: (a) If a >, and k, then (b) The function ζ() = University of Pittsburgh Preliminary Eam, 2009 n=2 Problem 7. Prove that the function log k n n a <., >, is infinitely differentiable in (, ). n University of Pittsburgh Preliminary Eam, 2009

4 4 INSTRUCTOR: CEZAR LUPU is continuous on R. f() = ( n n! ) 2 University of Pittsburgh Preliminary Eamination, 2005 Problem 8. (a) Give sufficient conditions under which the series f n () to be differentiated term by term on a bounded interval I R. (b) Can we differentiate arctan term by term? n2 University of Pittsburgh Preliminary Eam, 2006 Problem 9. For n N and R +, define f () =, f n+ () = + f n (), n. Prove that f n converges uniformly on every interval [a, b] where 0 < a < b <. Problem 20. Determine and lim t 0 Ohio State University Qualifying Eam, 999 n 2 + t 2, lim t n 2 + t 2. Problem 2. Prove that the two series Ohio State University Qualifying Eam, 995 and c n n, n log nc n n+3 have the same radius of convergence. University of Pittsburgh Preliminary Eam, 2009

5 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (SEQUENCES OF FUNCTIONS, SERIES n n Problem 22. Prove that n + defines a continuous function on [0, ). is not uniformly convergent on [0, ), but it University of Pittsburgh Preliminary Eam, 203 Problem 23. Prove that the series f() = sin n n 5/2 converges for all R and that f() is a continuous function on R with continuous derivative. UCLA Basic Eamination, 2006 Problem 24. Let g be a continuous function on R with g(0) = 0 and let g be bounded on R, that is (a) Show that the series sup{ g () : R} = M <. converges for all R and that its sum is a continuous function on R. (b) Is f differentiable on R? f() = ( ) n g n ( ) n g n Problem 25. Let I be the interval [0, ). For n N and t I, let f n (t) = sin((t + 4n 2 π 2 ) /2 ), n. (i) Show that the sequence (f n ) is equicontinuous on I. (ii) Show that (f n ) does not contain a subsequence which is uniformly convergent on I. University of Pittsburgh Preliminary Eam, 200 Problem 26. Let f n () = e n ( + n ) n 2, defined for any real and for any n N.

6 6 INSTRUCTOR: CEZAR LUPU (a) Prove that there is a function f : R R, with lim n f n () = f(), for each real number and determine this function eplicitly. (b) Is the convergence of f n () to f() uniform? University of Pittsburgh Preliminary Eam, 204 Problem 27. Prove the identity, valid for any real, with < : ( ) = k k. 2 Now let f : ( 3, 3) R be given by the series, valid for any real with < 3: Prove that f () f() = k= ( ) k. ( ) k + 4 k= 3, for any real number, such that 0 3. (3 ) 2 University of Pittsburgh Preliminary Eam, 204 Problem 28. Let f n : [0, ) R be the function defined by f n () = n k= k + k. (a) Prove that f n converges to a function f : [0, ) R. (b) Prove that for every 0 < a < the convergence is uniform on [0, a]. (c) Prove that f is differentiable on (0, ). University of Lincoln-Nebraska Qualifying Eam, 202 Problem 29. Let f n () = ( 2 ) 2n, for n = 0,, 2,... and for [, ]. (a) Show that the series the limit. (b) Show that the series f n () converges pointwise but not uniformly, and find ( ) n f n () converges uniformly on [, ]. University of Lincoln-Nebraska Qualifying Eam, 2006 Problem 30. (a) Consider the functions f n () =, for 0. Show that + n 3 2 f n converge pointwise to zero on [0, ). For which a 0, if any, does f n converge uniformly on [a, )? (b) Suppose that a sequence of uniformly continuous functions f n : [0, ) R converge uniformly to a function f : [0, ) R. Prove that f is uniformly continuous. n2

7 WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (SEQUENCES OF FUNCTIONS, SERIES University of Lincoln-Nebraska Qualifying Eam, 2004 Problem 3. Let f n : X R, n =, 2,... be a sequence of continuous functions on a metric space X such that the series f n() converges for all X and ( sup X f n () 2 ) /2 <. Prove that if a sequence of real numbers c n, n =, 2,... satisfies c2 n <, then the series c n f n () converges everywhere to a continuous function. University of Pittsburgh Preliminary Eam, 205