Prove the following does not converge uniformly in [0,pi] but does converge uniformly in each closed subinterval [a,b] of (0, pi) Σ h=1 coshx log(h² +et)
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![Prove the following does not converge
uniformly in [0,pi] but does converge uniformly
in each closed subinterval [a,b] of (0, pi)
h=1
coshx
log(h² +e)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e2f8657-770b-46ba-9ea3-6c2ee9e94eec%2F2d0a1b60-c592-4374-bbb9-e3cda05beb49%2F1y7txmg_processed.jpeg&w=3840&q=75)
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- Determine if the following improper integral converge or diverge. If it covers to what?Find all real numbers p for which the improper integral ∫∞1t−pdt∫1∞t−pdt converges. For what values of p does it diverge?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)?
- Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.Find the radius of convergence for zn/n , n = 1 to infinity.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,∞).
- Suppose that ∞ n = 0 anxn converges to a function y such that y'' − 2y' + y = 0 where y(0) = 0 and y'(0) = 1. Find a formula that relates an + 2, an + 1, and(a) Use the fact that the Taylor series for sin(x) and cos(x) converge uniformly on any compact set to prove WAT under the added assumption that [a, b] is [0, π]. (b) Show how the case for an arbitrary interval [a, b] follows from this one.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.