1. Consider a sequence of functions defined on R bytr$(2)=n(i) Sketch the graphs of f1, f2, f3 and f₁ on the same set of axes.(ii) Find the pointwise limit of the sequence.(iii) Show that the convergence is uniform on any bounded subset of R.

The given sequence of function is, fnx=x2+nxn. (i) To Sketch: graphs of f1,f2, f3, and f4 on same…

Answered: 1. Consider a sequence of functions…

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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4. State the dominated convergence theorem and use it to find the limit when n�� ∞ of the following sequences : (a) f(t)dt, where ƒ is a continuous function on (0, 1), ) 1₁ ² 1 (1+t") (b) dt.
4. Let fn(x) = x - x". Prove that fn converges pointwise on [0, 1] and find the limit function. Is the convergence uniform on [0, 1]? Is the convergence uniform on [0, 1)?
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?
(5) Consider the sequence defined by -( + 2) for n21 S1 = 1 Sn+1 = (Sn 4 (a) Prove by induction for all n E N (b) Prove the sequence is monotone. (c) Use the previous two parts to explain whether or not the limit of s, exists. If the limit exists, find the limit.
Construct a sequence of continuous functions fn : R → R such that 0 < fn s 1, limn0 Jo fndx = 0 but that the sequence fn() converges for no x € [0, 1].
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,…
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3. Let an = (3n +4n)¹/n (a) Prove that for all n € N, 4 < an = 4(1 + (²)n)¹/n < 4.2¹/n (b) Prove that the sequence {an}=1 is convergent and find its limit.
4. Let f be differentiable on R with a = sup{|ƒ'(x)| : x € R} < 1. f(sn-1) for n ≥ 1. Prove that (i) Pick a number so € R and inductively define sn = {n} is a convergent sequence in R. (ii) Show that f has a fixed point, that is, f(s) = s for some s E R.
Use Monotone Convergence Theorem to prove that (Sn) converges and find its limit.

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1. Consider a sequence of functions defined on R by giả th n f(z)= (i) Sketch the graphs of f1, f2. f3 and fa on the same set of axes. (ii) Find the pointwise limit of the sequence. (iii) Show that the convergence is uniform on any bounded subset of R.