Homework Answers
Add Answer to:
EX.3: Find a counterexample to the following statement: If {fr} converges uniformly to f on an...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that the function f(x) is also differentiable. (All of these conditions are assumed to be true on a bounded, closed interval [a, b].) Prove or disprove: lim as n goes to infinity fn'(x) = f'(x)
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
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].
10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additiona...
10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additional assumption that each fis continuous on I la, b).(Hint: For x in la, b.o)If f g uniformly on [a, b], then Theorem 8.3 implies that im f.(x) f (x8. It follows that frpuintwise on la, b), where F(x) -lim frCro) + .By Theorem 6.12, F()-x) on la,b). Now show that f uniformly on la, b].] F heorem 8.4 Suppose that neN is a...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at le...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at least punctually in its domain? b)If {fn (2.)) is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Is {fn (x)) a succes- sion of functions uniformly bounded?
For the following statements give a counterexample or demonstrate them: a)If fn (x) is...
= 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...
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...
Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly...
Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly to f :G+C, and y C G is a piecewise smooth path. Then lim n-00 $. fn = $. . 7.23. Let fn(x) = n2x e-nx. (a) Show that limn400 fn(x) = 0 for all x > 0. (Hint: Treat x = O as a special case; for x > 0 you can use L'Hôspital's rule (Theorem A.11) — but remember that n is...
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers....
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
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].
10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additional assumption that each fis continuous on I la, b).(Hint: For x in la, b.o)If f g uniformly on [a, b], then Theorem 8.3 implies that im f.(x) f (x8. It follows that frpuintwise on la, b), where F(x) -lim frCro) + .By Theorem 6.12, F()-x) on la,b). Now show that f uniformly on la, b].] F heorem 8.4 Suppose that neN is a...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at least punctually in its domain? b)If {fn (2.)) is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Is {fn (x)) a succes- sion of functions uniformly bounded?
For the following statements give a counterexample or demonstrate them: a)If fn (x) is...
= 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...
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...
Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly to f :G+C, and y C G is a piecewise smooth path. Then lim n-00 $. fn = $. . 7.23. Let fn(x) = n2x e-nx. (a) Show that limn400 fn(x) = 0 for all x > 0. (Hint: Treat x = O as a special case; for x > 0 you can use L'Hôspital's rule (Theorem A.11) — but remember that n is...
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
Most questions answered within 3 hours.
-
You follow the lab procedure and at the end of Part 4C, you used
an average...
asked 4 minutes ago -
Write 250-450 words, about the following: • Analyze how a
servant leader in your organization or...
asked 7 minutes ago -
3. American and Japanese workers can each produce 4 cars a
year. An American worker can...
asked 10 minutes ago -
On October 1, 20X1, a company purchased a piece of land by
agreeing to pay the...
asked 22 minutes ago -
Jamie is doing a survey at her school about whether the students
feel the cafeteria food...
asked 27 minutes ago -
What are the roles of 3' and 5' untranslated regions in
mRNAs?
asked 29 minutes ago -
A domestic television receives antenna delivers a sky noise
power of -105 dBm to a matched...
asked 27 minutes ago -
What do you think are the chemical reactions involved in the
breathalyzer tests administered to potentially...
asked 43 minutes ago -
A random sample of 120 observations produces a mean of
38.2 from a population with a...
asked 49 minutes ago -
The average area of land burned by forest fires every year in
Canada is 2.5 million...
asked 51 minutes ago -
36. The basic difference between government bureaus and market
firms is that bureaus:
a. can...
asked 51 minutes ago -
java
- A recursive method findR() to search for a number in list
(ArrayList)
- A...
asked 1 hour ago