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Homework Answers
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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...
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...
(4) Let(an}n=o be a sequence in C. Define R-i-lim suplanlì/n. Recall that R e [0,x] o0 is the rad...
(4) Let(an}n=o be a sequence in C. Define R-i-lim suplanlì/n. Recall that R e [0,x] o0 is the radius of convergence of the power series Σ a (z 20)" Assume that R > 0 (a) Prove that if 0 < ρ < R, then the power series converges uniformly on the closed (b) Prove that the power series converges uniformly on any compact subset of the disk Ix - xo< R
(4) Let(an}n=o be a sequence in C. Define R-i-lim...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Sho...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all inte...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
= 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...
This assignment asks you to prove the following Proposition 1 Let {n} and {n} are two...
This assignment asks you to prove the following Proposition 1 Let {n} and {n} are two sequences of real numbers and L is a number such that (1.a) un → 0, and (1.b) V EN, -L Swn. We illustrate the proposition. To begin, one can check from the definition that 1/n 0. This fact, plus the arithinetic rules of convergence, generate a large family of sequences known to converge to 0. For example, 11n +7 1 11 +7 3n2 -...
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
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i...
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i.e., f(-x) = -f(x)) means we can avoid doing any integrals for the moment and just appeal to a symmetry argument to conclude T f (x) cos(nar)dx 0 and an f(x)dax = 0 ao -- T 27T -T for all n 1. We can also simplify the integral for bn by...
a) Show that the series CO (e n 0 n 0 on the interval 10, co...
a) Show that the series CO (e n 0 n 0 on the interval 10, co towards the function Converges pointwise 1 t e]0, co[ f(t) 1 — е- b) Show that the series CO пе-nt п3D0 converges uniformly for t in the interval [b, o for every constant b > 0. Let CO ne nt t> 0, s(t) n 1 be the sum function of the series. J0, co[ d) Show thatf'(t) = -s(t) for all t > 0...
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...
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...
(4) Let(an}n=o be a sequence in C. Define R-i-lim suplanlì/n. Recall that R e [0,x] o0 is the radius of convergence of the power series Σ a (z 20)" Assume that R > 0 (a) Prove that if 0 < ρ < R, then the power series converges uniformly on the closed (b) Prove that the power series converges uniformly on any compact subset of the disk Ix - xo< R
(4) Let(an}n=o be a sequence in C. Define R-i-lim...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
= 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...
This assignment asks you to prove the following Proposition 1 Let {n} and {n} are two sequences of real numbers and L is a number such that (1.a) un → 0, and (1.b) V EN, -L Swn. We illustrate the proposition. To begin, one can check from the definition that 1/n 0. This fact, plus the arithinetic rules of convergence, generate a large family of sequences known to converge to 0. For example, 11n +7 1 11 +7 3n2 -...
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
Example 8.5.1. Let if 0< x< T if 0 or r? -1 if -т <т < 0. 1 f(x)= 0 _ The fact that f is an odd function (i.e., f(-x) = -f(x)) means we can avoid doing any integrals for the moment and just appeal to a symmetry argument to conclude T f (x) cos(nar)dx 0 and an f(x)dax = 0 ao -- T 27T -T for all n 1. We can also simplify the integral for bn by...
a) Show that the series CO (e n 0 n 0 on the interval 10, co towards the function Converges pointwise 1 t e]0, co[ f(t) 1 — е- b) Show that the series CO пе-nt п3D0 converges uniformly for t in the interval [b, o for every constant b > 0. Let CO ne nt t> 0, s(t) n 1 be the sum function of the series. J0, co[ d) Show thatf'(t) = -s(t) for all t > 0...
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