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2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr...
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10...
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...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
= 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...
1. (40pts) Let 8 >0 and hn: (8,2 - 8] -R be given by cos(n) hn...
1. (40pts) Let 8 >0 and hn: (8,2 - 8] -R be given by cos(n) hn (x) 72 Use Dirichlet's Test to show that the series hn converges uniformly on (8,27 - 8). That is, please solve the following problems: la. (10 pts) Let 9n (x) = . * € (8,27 - 8). Show that In - g uniformly, where g(x) = 0, for all 2 € (5,2 - 8) and 9n+1 () S (x). for all n e N...
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.
6. (10 points) Let fn: R+R be defined as fn(x) = (sin(ru) 2 Show fn converges...
6. (10 points) Let fn: R+R be defined as fn(x) = (sin(ru) 2 Show fn converges pointwise but not uniformly on R.
la b c Given that d e f =-1 , find 18 hill | 2a 2b...
la b c Given that d e f =-1 , find 18 hill | 2a 2b 2c -2d -2e | 6g 6h 6i || 2a 2b 2c | |-20 -2e -2f= | 6g 6h 6i exact number, no tolerance
5. Let fn(x) = x"/n on [0, 1]. Show that (fr)nen converges uniformly to a differentiable...
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].
prove that A is non singular 5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let...
prove that A is non singular
5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?
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...
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...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.
= 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...
1. (40pts) Let 8 >0 and hn: (8,2 - 8] -R be given by cos(n) hn (x) 72 Use Dirichlet's Test to show that the series hn converges uniformly on (8,27 - 8). That is, please solve the following problems: la. (10 pts) Let 9n (x) = . * € (8,27 - 8). Show that In - g uniformly, where g(x) = 0, for all 2 € (5,2 - 8) and 9n+1 () S (x). for all n e N...
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.
6. (10 points) Let fn: R+R be defined as fn(x) = (sin(ru) 2 Show fn converges pointwise but not uniformly on R.
la b c Given that d e f =-1 , find 18 hill | 2a 2b 2c -2d -2e | 6g 6h 6i || 2a 2b 2c | |-20 -2e -2f= | 6g 6h 6i exact number, no tolerance
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].
prove that A is non singular
5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?
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...
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