Please help with this math question![24.15 Let fn (2) = the for x € (0,0). (a) Find f(x) = lim fn(x). (b) Does fn + f uniformly on [0, 1]? Justify. (c) Does fn +](http://img.homeworklib.com/questions/92008800-b2ce-11ec-8b3a-b7da8de9c6cc.png?x-oss-process=image/resize,w_560)
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24.15 Let fn (2) = the for x € (0,0)....
help me solve 24.11 24.11 Let fn(x) = x and gn(x) = 7 for all x...
help me solve 24.11
24.11 Let fn(x) = x and gn(x) = 7 for all x € R. Let f(x) = x and g(x) = 0 for x E R. (a) Observe fn + f uniformly on R (obvious!) and In + g uniformly on R (almost obvious]. (b) Observe the sequence (fron) does not converge uniformly to fg on R. Compare Exercise 24.2. 24.2 For x € [0,00), let fn(x) = (a) Find f(x) = lim fn(x). (b) Determine...
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y....
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2)...
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.
(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...
(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
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m =...
= 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...
QUESTION 2 -2 if -π,-1/2) x Let f(x) = if XE [-1/2, 1 /2) x 2 if xE1/2, T] If FN(x)is the partial...
QUESTION 2 -2 if -π,-1/2) x Let f(x) = if XE [-1/2, 1 /2) x 2 if xE1/2, T] If FN(x)is the partial sum of the Fourier series off(x), then give lim FN(1/2h
QUESTION 2 -2 if -π,-1/2) x Let f(x) = if XE [-1/2, 1 /2) x 2 if xE1/2, T] If FN(x)is the partial sum of the Fourier series off(x), then give lim FN(1/2h
Proof Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous...
Proof
Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S
Theorem 65.6 (a generalization of Dini's theorem) Let...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx)...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
lim 00 lim 3a. (10 pts) Let Sn: 10,1] - R be defined by fn (x)...
lim 00 lim 3a. (10 pts) Let Sn: 10,1] - R be defined by fn (x) = nºx (1 - )". Is is true that ( sn (2) de - 1 In (x) dx. 3b. (10 pts) Let fn (x) = 1+2+3* € (0,1). Find I In (2) de Justify your answer. lim
part (c) 7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all...
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
help me solve 24.11
24.11 Let fn(x) = x and gn(x) = 7 for all x € R. Let f(x) = x and g(x) = 0 for x E R. (a) Observe fn + f uniformly on R (obvious!) and In + g uniformly on R (almost obvious]. (b) Observe the sequence (fron) does not converge uniformly to fg on R. Compare Exercise 24.2. 24.2 For x € [0,00), let fn(x) = (a) Find f(x) = lim fn(x). (b) Determine...
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.
(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
= 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...
QUESTION 2 -2 if -π,-1/2) x Let f(x) = if XE [-1/2, 1 /2) x 2 if xE1/2, T] If FN(x)is the partial sum of the Fourier series off(x), then give lim FN(1/2h
QUESTION 2 -2 if -π,-1/2) x Let f(x) = if XE [-1/2, 1 /2) x 2 if xE1/2, T] If FN(x)is the partial sum of the Fourier series off(x), then give lim FN(1/2h
Proof
Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S
Theorem 65.6 (a generalization of Dini's theorem) Let...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
lim 00 lim 3a. (10 pts) Let Sn: 10,1] - R be defined by fn (x) = nºx (1 - )". Is is true that ( sn (2) de - 1 In (x) dx. 3b. (10 pts) Let fn (x) = 1+2+3* € (0,1). Find I In (2) de Justify your answer. lim
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
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