heorem 8.4 Suppose that neN is a sequence of differentiable functions defined on a bounded interval/and that (fconverges for some point (fn)"EN converges uniformly on I , then (fn),eN COnverges uniformly on / to a differentiable function F and F'(a)lim,(x) for all x in l
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10. Use the Fundamental Theorem of Calculus to provide a proof of Theorem 8.4 under the additiona...
Use the Mean Value Theorem to supply a proof for Theorem 6.3.2. To get started, observe...
Use the Mean Value Theorem to supply a proof for Theorem 6.3.2. To get started, observe that the triangle inequality implies that, for any x є [a,b] and m, n є N Theorem 6.3.2. Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b, and assume (%) converges uniformly on [a, b. If there erists a point to E [a, b] where n(o) is convergent, then (f) converges uni- formly on [a,
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...
Analysis: Give two examples where if fn does not converge to f uniformly on E, but does converge ...
Analysis:
Give two examples where if fn does not converge to f uniformly
on E, but does converge to f pointwise on E, then the following two
theorems do not hold. Write clearly and explain and proof your
claims.
711 Theorem Suppose fn→f uniformly on a set E in a metric space. Let x be a limit point of E, and suppose that (15) Then (A,) converges, and (16) lim f()im A In other words, the conclusion is that lim...
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...
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it...
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r <...
Hint: Use the fundamental theorem of arithmetic. 15. Theorem 14.5 implies that Nx N is countably infinite. Construct an alternate proof of this fact by showing that the function ф : N x N 2n-1(...
Hint: Use the fundamental theorem of arithmetic.
15. Theorem 14.5 implies that Nx N is countably infinite. Construct an alternate proof of this fact by showing that the function ф : N x N 2n-1(2m-1) is bijective. N defined as ф(m,n) It is also true that the Cartesian product of two countably infinite sets is itself countably infinite, as our next theorem states. Theorem 14.5 If A and B are both countably infinite, then so is A x B. Proof....
PLEASE use the THEORY below to give PROOF STEP BY STEP. This is an analysis class. Thank you. application of power series\Weierstrass M-test\term by term differentiability of power series sequence an...
PLEASE use the THEORY below to
give PROOF STEP BY STEP. This is an analysis class. Thank
you.
application of power series\Weierstrass M-test\term by term
differentiability of power series
sequence and series of function: pointwise and the theorem of
uniform convergence
which function is integrable: continuous and monotone
Fri 19 Apr: The Fundamental Theorem of Calculus. (§7.5.)
Wed 17 Apr: Example (∫10x2dx=1/3∫01x2dx=1/3). Basic properties
of the integral. (mostly Theorem 7.4.2.)
Fri 12 Apr: More on integrability, basic properties of the...
Problem 3. Prove Theorem 1 as tollows [Assume all conditions of the Theorem are met. In many part...
Problem 3. Prove Theorem 1 as tollows [Assume all conditions of the Theorem are met. In many parts, it will be useful to consider the sign of the right side of the formula-positive or negative- ad to write the appropriate inequality] (a) Since f"(x) exists on [a, brx) is continuous on [a, b) and differentiable on (a,b), soMean Value Thorem applies to f,on this interval. Apply MVTtof"m[x,y], wherc α zcysb. to show that ry)2 f,(x), İ.e. that f, is increasing...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vec...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vector field F(x.y)-(r-yi+r+y)j and the circle C: r y-9. Verify Green's Theorem by calculating the outward flux of F across C (12 points) Find the absolute extreme values of the function .-2-4--3 on the closed triangular region in the xy-plane bounded by the lines x...
Problem 11.11 I have included a picture of the question (and the referenced problem 11.5), followed...
Problem 11.11
I have included a picture of the question (and the referenced
problem 11.5), followed by definitions and theorems so you're able
to use this books particular language. The information I include
ranges from basic definitions to the fundamental theorems of
calculus.
Problem 11.11. Show, if f : [0,1] → R is bounded and the lower integral of f is positive, then there is an open interval on which f > 0. (Compare with problems 11.5 above and Problem...
Use the Mean Value Theorem to supply a proof for Theorem 6.3.2. To get started, observe that the triangle inequality implies that, for any x є [a,b] and m, n є N Theorem 6.3.2. Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b, and assume (%) converges uniformly on [a, b. If there erists a point to E [a, b] where n(o) is convergent, then (f) converges uni- formly on [a,
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...
Analysis:
Give two examples where if fn does not converge to f uniformly
on E, but does converge to f pointwise on E, then the following two
theorems do not hold. Write clearly and explain and proof your
claims.
711 Theorem Suppose fn→f uniformly on a set E in a metric space. Let x be a limit point of E, and suppose that (15) Then (A,) converges, and (16) lim f()im A In other words, the conclusion is that lim...
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...
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r <...
Hint: Use the fundamental theorem of arithmetic.
15. Theorem 14.5 implies that Nx N is countably infinite. Construct an alternate proof of this fact by showing that the function ф : N x N 2n-1(2m-1) is bijective. N defined as ф(m,n) It is also true that the Cartesian product of two countably infinite sets is itself countably infinite, as our next theorem states. Theorem 14.5 If A and B are both countably infinite, then so is A x B. Proof....
PLEASE use the THEORY below to
give PROOF STEP BY STEP. This is an analysis class. Thank
you.
application of power series\Weierstrass M-test\term by term
differentiability of power series
sequence and series of function: pointwise and the theorem of
uniform convergence
which function is integrable: continuous and monotone
Fri 19 Apr: The Fundamental Theorem of Calculus. (§7.5.)
Wed 17 Apr: Example (∫10x2dx=1/3∫01x2dx=1/3). Basic properties
of the integral. (mostly Theorem 7.4.2.)
Fri 12 Apr: More on integrability, basic properties of the...
Problem 3. Prove Theorem 1 as tollows [Assume all conditions of the Theorem are met. In many parts, it will be useful to consider the sign of the right side of the formula-positive or negative- ad to write the appropriate inequality] (a) Since f"(x) exists on [a, brx) is continuous on [a, b) and differentiable on (a,b), soMean Value Thorem applies to f,on this interval. Apply MVTtof"m[x,y], wherc α zcysb. to show that ry)2 f,(x), İ.e. that f, is increasing...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vector field F(x.y)-(r-yi+r+y)j and the circle C: r y-9. Verify Green's Theorem by calculating the outward flux of F across C (12 points) Find the absolute extreme values of the function .-2-4--3 on the closed triangular region in the xy-plane bounded by the lines x...
Problem 11.11
I have included a picture of the question (and the referenced
problem 11.5), followed by definitions and theorems so you're able
to use this books particular language. The information I include
ranges from basic definitions to the fundamental theorems of
calculus.
Problem 11.11. Show, if f : [0,1] → R is bounded and the lower integral of f is positive, then there is an open interval on which f > 0. (Compare with problems 11.5 above and Problem...
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