A sequence {ann21 contained in the interval [0, 1] is uniformly distributedeif and only iflim — [ f(a) = f f(x) dxnào nk=1-holds for any continuous function f(x) defined on [0, 1].

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Answered: A sequence {ann21 contained in the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose {f} is a sequence of functions, differentiable on [a, b] and such that {fn (xo)} converges for some point xo on [a, b]. If {f'n} converges uniformly on [a, b], then {f} converges uniformly on [a, b], to a function f, and (a ≤ x ≤ b). f'(x) = lim f'(x) n2-00 Can we replace [a, b] by [0, ∞o)? Why?
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an Assume f(x) = ) anx" converges on (-R, R). Show that F(x) = > n+1 is defined n+1 n=0 n=0 on (–R, R) and satisfies F' = f on (-R, R).
9. Let (x) and (yn) be two sequences converges to c with x + yn for all n. Assume that fis defined for all real numbers and that f'(c) exists. Prove that [10w-₂)} converges to f'(c).
In general, if f(x) Tn O O [ f(x) dx > Tn Since, M is the area under ---Select--- ✓to the graph of f(x), then f"(x) OM,> [ f(x) dx < [°r(x) dx OM₁ < f(x) dx OMn< f(x) dx Thus, T< < [*r(x) dx < M₁² ✓ the graph. Therefore, we have which of the following?
18
4. If (x,) is Cauchy and has a convergent subsequnece (x), then prove that (r,) is convergent and has the same limit as (xn).
3. Let {gn} be a sequence of integrable functions such that gn →g a.e. with g integrable. Show that lim f [gn – g| = 0 if and only if lim f \gn| = S \g].
Sketch the graph of a function continuous on the given interval that satisfies the following condition. ƒ is continuous on (- ∞, ∞); ƒ'(x) < 0 and ƒ"(x) < 0 on(- ∞, 0); ƒ'(x) > 0 and ƒ"(x) > 0 on (0, ∞).
Prove the following
J let 1₁ = √6, AM+₁ Лин si = √6 +^₂; 1=1,2,. Votons Prove that (on) is convergent and find its limit
Let fi, f2,..., fkr 8ı, 82,..., g& be asymptotically nonnegative functions. Assume that fi(n) = O(g;(n)) for each i = 1,2,. ,k, prove that k k 1. ) fi(n) = 0(max{g;(n)} \1gisk 2. || f(n) = 0 8i(n) i=1 i=1 i=1

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A sequence {ann21 contained in the interval [0, 1] is uniformly distributede if and only if lim -- [ f(ak) = Σ I' f(x) dx n—c n k=1 -holds for any continuous function f(x) defined on [0, 1].