4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform? 4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn}...
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but (Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform 3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform
2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr = 2Ě (2n-1) 2d. (10 pts) Let 0 < < be given. Show that f given in 2b) is differentiable at each 1 € (5,27 - 8). Find f' (1). Hint: Use Problem 1 and the following formula In 2 (-1)"-1 Σ 7 n=1 2. (40 pts) Let fn: R → R be given by fn (x) = sin (nx) 3 ηε Ν. n2...
5. Let fn(x) = x"/n on [0, 1]. Show that (fr)nen converges uniformly to a differentiable function on [0, 1], but (f%) does not converge uniformly neN on [0, 1].
For n ∈ N let fn : [0, 1] → R be given by fn(x) = sin((1 + n)x) / ( 1 + n ) ^(1/2) . Prove that {fn} is equicontinuous on [0, 1].
Many thanks!! (a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...