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4. Consider the sequence of functions fu(x) = 1 timer defined on the whole real line...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the in...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function wit...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1]. Find...
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1].
Find a function h in B([0,1]) such that
du(f,h) = du(f,g) = du(g,h).
(3 problems)
13.2.6. Problem. Given in each of the following is the nth term of a sequence of real valued functions defined on (0, 1]. Which of these converge pointwise on (0, 1]? For which is the convergence uniform? (a) a z" (b) z+ nr. (c) a+ re-na 13.2.7. Problem. Given in...
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise...
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose the sequence {f) converges uniformly on every compact subset of (a, b). Prove thatf is differen- tiable on (a, b) and that f'(x) = lim f(x) for all x E (a, b).
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose...
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 poi...
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
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}...
Please answer it step by step and Question 2. uniformly converge is defined by *f=0* clear...
Please answer it step by step and Question 2. uniformly
converge is defined by *f=0* clear handwritten,
please, also, beware that for the x you have 2 conditions , such as
x>n and 0<=x<=n
1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval -00 < x < oo that approach zero at least as quickly as xas +00 (a) (4 points) Show that the operator B=x+in is Hermitian when acting upon {f(x)}. (b) (4 points) Show that A = x + (4) is not Hermitian and determine the operator At. Determine C = AtA and show that it is Hermitian. (c) (2 points) What well-known problem in...
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the...
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1 STO X STO 3-X ENTER, ENTER ENTER ENTER. 4-x Apparently the sequence diverges / converges to b. State the MONOTONE CONVERGENCE THEOREM: c. Use induction to show that (x) is decreasing for all n when x, 3 d. Use induction to show that (x.) is bounded below by 0 when x,- 3. e. Conclude from (b-d): d. To...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
13.1.11. Problem. Let f(x) = x and g(x) = 0 for all x ∈ [0,1].
Find a function h in B([0,1]) such that
du(f,h) = du(f,g) = du(g,h).
(3 problems)
13.2.6. Problem. Given in each of the following is the nth term of a sequence of real valued functions defined on (0, 1]. Which of these converge pointwise on (0, 1]? For which is the convergence uniform? (a) a z" (b) z+ nr. (c) a+ re-na 13.2.7. Problem. Given in...
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose the sequence {f) converges uniformly on every compact subset of (a, b). Prove thatf is differen- tiable on (a, b) and that f'(x) = lim f(x) for all x E (a, b).
4. Let {S.} be a sequence of differentiable real-valued functions on (a, b) that converges pointwise to a function f on (a, b). Suppose...
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
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}...
Please answer it step by step and Question 2. uniformly
converge is defined by *f=0* clear handwritten,
please, also, beware that for the x you have 2 conditions , such as
x>n and 0<=x<=n
1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
2. Consider the set of functions {f(x)} of the real variable x, defined on the interval -00 < x < oo that approach zero at least as quickly as xas +00 (a) (4 points) Show that the operator B=x+in is Hermitian when acting upon {f(x)}. (b) (4 points) Show that A = x + (4) is not Hermitian and determine the operator At. Determine C = AtA and show that it is Hermitian. (c) (2 points) What well-known problem in...
3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1 STO X STO 3-X ENTER, ENTER ENTER ENTER. 4-x Apparently the sequence diverges / converges to b. State the MONOTONE CONVERGENCE THEOREM: c. Use induction to show that (x) is decreasing for all n when x, 3 d. Use induction to show that (x.) is bounded below by 0 when x,- 3. e. Conclude from (b-d): d. To...
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