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2. (a) Suppose that f is Lebesgue integrable on R. Find the following limit: n sin(x/n f(x) dz. (b) Find the value of the limit in the special case: linn onsin(x/n) n→oo/.oo X(X2 + 1) dx. 2....
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue...
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebes...
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length 2π Hint: I is contained in two consecutive...
Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length 2π Hint: I is contained in two consecutive intervals of the form (kT, (k+2)π)
Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length...
(4) Define the function f : R -> R* by ,--1/2 f(x) x< 0. +oo, |(a) Prove that f is measurable (with respect to th...
(4) Define the function f : R -> R* by ,--1/2 f(x) x< 0. +oo, |(a) Prove that f is measurable (with respect to the Lebesgue measurable sets). (b) Prove that f is integrable on I 0, 1and compute the value of = f du
(4) Define the function f : R -> R* by ,--1/2 f(x) x
R such that f is integrable on every [a,b] (6) Suppose f is a function and...
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
Compute the following limits and justify the calculations: a. limo*(1 (/n))-n sin(x/n) dz. c. lim...
Compute the following limits and justify the calculations: a. limo*(1 (/n))-n sin(x/n) dz. c. limn-Joon sin(x/n)[x(1+x2)]-1 dr. d. limn→00 Jaon( 1 + n2x2)-1 d. (The answer depends on whether a > 0, . limn-oo a = 0, or a < 0, How does this accord with the various convergence theorems?)
Compute the following limits and justify the calculations: a. limo*(1 (/n))-n sin(x/n) dz. c. limn-Joon sin(x/n)[x(1+x2)]-1 dr. d. limn→00 Jaon( 1 + n2x2)-1 d. (The answer depends on whether a...
Suppose that f is bounded on a, b and that for any cE (a, b), f...
Suppose that f is bounded on a, b and that for any cE (a, b), f is integrable on [c, b (a) Prove that for every e> 0, there exists CE (a, b) so that f(x)(c-a) < € for all x [a,b]. (b) For any > 0, find a partition P of [a, b so that U,P)-J f(r)dz < j and s f(r)dz L(f, P) < Hint: Do this by choosing c carefully and extending a partition of [c, b...
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x)...
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
If f is integrable on [a, b], the following equation is correct. Integral^b_a f (x) dx...
If f is integrable on [a, b], the following equation is correct. Integral^b_a f (x) dx = lim_n rightarrow infinity Sigma^n _i = 1 f (x_i) Delta x, where Delta x = b - a/n and X_i = a + i delta x. Use the given form of the definition to evaluate the integral. integral^1_0 (2 - x^2) dx
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)...
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length 2π Hint: I is contained in two consecutive intervals of the form (kT, (k+2)π)
Suppose f is integrable on (-π, π] and extended to R by making it periodic of period 2π. Show that f(x) dx= | f(x)dz where I is any interval in R of length...
(4) Define the function f : R -> R* by ,--1/2 f(x) x< 0. +oo, |(a) Prove that f is measurable (with respect to the Lebesgue measurable sets). (b) Prove that f is integrable on I 0, 1and compute the value of = f du
(4) Define the function f : R -> R* by ,--1/2 f(x) x
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
Compute the following limits and justify the calculations: a. limo*(1 (/n))-n sin(x/n) dz. c. limn-Joon sin(x/n)[x(1+x2)]-1 dr. d. limn→00 Jaon( 1 + n2x2)-1 d. (The answer depends on whether a > 0, . limn-oo a = 0, or a < 0, How does this accord with the various convergence theorems?)
Compute the following limits and justify the calculations: a. limo*(1 (/n))-n sin(x/n) dz. c. limn-Joon sin(x/n)[x(1+x2)]-1 dr. d. limn→00 Jaon( 1 + n2x2)-1 d. (The answer depends on whether a...
Suppose that f is bounded on a, b and that for any cE (a, b), f is integrable on [c, b (a) Prove that for every e> 0, there exists CE (a, b) so that f(x)(c-a) < € for all x [a,b]. (b) For any > 0, find a partition P of [a, b so that U,P)-J f(r)dz < j and s f(r)dz L(f, P) < Hint: Do this by choosing c carefully and extending a partition of [c, b...
Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
If f is integrable on [a, b], the following equation is correct. Integral^b_a f (x) dx = lim_n rightarrow infinity Sigma^n _i = 1 f (x_i) Delta x, where Delta x = b - a/n and X_i = a + i delta x. Use the given form of the definition to evaluate the integral. integral^1_0 (2 - x^2) dx
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
Let {h} be a sequence ofRiemann integrable functions on [a,b], such that for each x, {h(x)) is a decreasing sequence. Suppose n) converges pointwise to a Riemann integrable function f Prove that f(x)dxf(x)dx. lim n00
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