2.60 Prove that fn(x) = - 0 pointwise on R but not uniformly on R. However, prove the convergence is uniform on (0, 1].
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![Prove that fn(x) = *
prove the convergence is uniform on (0, 1].
2.60
O pointwise on R but not uniformly on R. However,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe87a569b-7721-4104-9b29-fedcf0a2817b%2F4f4960f9-d7f0-4bec-b57f-919a382401a8%2Fastontp_processed.png&w=3840&q=75)
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- Let fn(x) = nx/1 + nx2 (d) Is the convergence uniform on (1,∞)?Suppose we have the sequence of functions $f_n(x)=x^n$ defined on $[0,1],$ and suppose $f_n\to f$ pointwise where $f(x)=0$ if $x\in(0,1]$ and $f(x)=1$ if $x=1.$ Prove that Uniform Convergence fails.Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.
- Let fn(x) = x^n for x ∈ [0,1]. check if it is pointwise convergence. Define where it becomes discontinuous.Assume that, for each n, fn is an integrable function on [a, b]. If (fn) → f uniformly on [a, b], prove that f is also integrable on this set. (We will see that this conclusion does not necessarily follow if the convergence is pointwise.)The series converges for every x in the half-open interval [−1, 1) but does not convergewhen x = 1. For a fixed x0 ∈ (−1, 1), explain how we can still use theWeierstrass M-Test to prove that f is continuous at x0.
- Is Uniform convergence of {fn(x)} is sufficient but not necessary to transmit continuity from the individual terms to the limit function ?Use the Cauchy Condensation test to prove that ∑ n = 2 to ∞ 1/( n (ln(n))^ p)) converges if p > 1 and diverges if p ≤ 1. (Make sure you verify that the hypothesis of the Cauchy Condensation test are met)Determine the convergence of the integralZ π2π3tan x(ln(cos x))2dx.
- Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as well. Is the claim true if we replace the assumption of uniform convergence with pintwise convergence.For any integer n ≥ 1 and any x ∈ (0,∞), define fn(x)= nx/(1+nx) (a) Let a > 0 be given. Prove that {fn} converges uniformly on the interval (a, ∞). (b) Prove that {fn} does not converge uniformly on (0,∞).Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly for x ∈ (0, +inf))? If yes, what is the uniform norm of fn(x)-f(x) on (0, +inf)?