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2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10...
2c. (10 pts) Show that f given in 2b) is intergrable and [ 1 (2) dr...
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...
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...
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.
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].
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.
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 o...
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].
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.
= 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...
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
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}...
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}...
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...
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...
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.
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].
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.
= 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...
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
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}...
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