1. Prove that the series7n(a) converges uniformly on every bounded interval(b) does not converge absolutely for any value of r2. Suppose f is a real continuous function on R, n (t) (nt) and { n :n N) is equicontinuous on3. Suppose is an equicontinuous sequence of functions on a compact set K, and {fn) converges. Let K be a compact metric space, and S CC(K), where C (K) is the space of all complex continuous0,1. What conclusion can you draw about f?pointwise on K. Show that {fn converges uniformlyfunctions on K, equipped with the metric d ( .9) = max EK | (x)-g(z) . Suppose that s is closedand pointwise bounded. Show that S is compact5. Let {fn be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, andput岔Fn (x)/fn (t) dt for a ssb0Show that there exists a subsequence {Fn^1 which converges uniformly on [a, b

As per norms . three questions ,1a, 1b and 2 are answered. To analyze various aspects of convergence…

Answered: 1. Prove that the series 7n (a)…

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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We define an arithmetic sequence of functions: fn: [1, 00) → R recursively as: f1(x) = Vx, fn + 1 (x) = /x + fn (x) ,Vx € [1,0) Vx E [1, 0) a- Prove that the sequence converges pointwise in [1, 00) and find its limit. b- Prove that for every a > 1 the sequence converges uniformly in [1, a]
1. Let fgng be a sequence of integrable functions which converges a.e. to an integrable function g. Let ffng be a sequence of measurable functions such that jf j g and ff g converges
Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is uniformly convergent on R.
Provide an example or give a reason why the request is impossible. (a) A sequence (fn) → f pointwise, where each fn has at most a finite numberof discontinuities but f is not integrable. (b) A sequence (gn) → g uniformly where each gn has at most a finite number of discontinuities and g is not integrable. (c) A sequence (hn) → h uniformly where each hn is not integrable but h is integrable.
3. Consider the sequence of functions on R given by fn(x) = (x - 1/n)². Prove that it converges pointwise and find the limit function. Is the convergence uniform on R? Is the convergence uniform on bounded intervals?
Q1: State whether the following statements are true or false: 1. Every Cauchy sequence is a convergent sequence in any matric space, but the converse is not true in general. 2. The convergence or divergence of a sequence depends only on the underlying space and not on the metric. 3. If f:M - S is a continuous function between matric spaces (M, dm) and (S, ds) at a point pEM, then for every closed set A in S, f-"(A) need not to be closed in M. 4. If f:M - S is a continuous function between matric spaces (M, dm) and (S, ds) at an isolated point p e M, then lim,-p f(x) = f(P). S. the sequence () is a Cauchy sequence in ([0, 00). I 1), but it is not convergent. 6. The subspace Q of rational numbers of R, forms a complete subspace of R. 7. Every finite subset of any matric space is complete. 8. If f:R -R be a continuous function on (a, b), then f[a, b] is bounded set but not closed in R. 9. Let f:M S be a function from a metric spaces (M, dm) and (S, ds). If f is uniformly continuous on M,…
IF in a normed space X, the absolute convergence of every series always implies convergence of the series show that X is a Banach space.
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,∞).
1. For each nЄ Z+ let fn: RR be the function nx f(x) = nx+1 (a) Prove that (fn) converges pointwise to some function f: RR. (b) Prove that (fn) does not converge uniformly.
Show that on finite measure spaces, LP and almost everywhere convergence are independent. That is, for any p it is possible that a sequence of functions converges in LP but not a.e. and it is also possible that it converges a.e. but not in LP.
1. Let c > 0 be fixed and define the sequence {an} by setting an (an? + 3c) За,2 + с 2 а1 > 0, An+1 for n E N. Determine all a1 for which the sequence converges and in such a case find its limit. 2. Prove that if f is a bounded function on [0,1] satisfying f(ax) bf (x) for 0 1, then lim f(x) = f(0). | 1 3. Suppose that f:[1, c0) → R is uniformly continuous. Prove that there is a positive M such that- If (x)| 1. 4. A function f: [0,1] → R satisfies f (0) 0, and there exists a function g continuous on [0,1] and such that f + g is decreasing. Prove that the equation f (x) = O has a solution in the open interval (0,1).

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