7. Consider the set Seq[ℕ] of all the infinite sequences (a_k)_{k ∈ ℕ} = (a_0, a_1, a_2, a_3, ...) of natural numbers. A sequence (a_k)_{k ∈ ℕ} is constant if there is some number n ∈ ℕ so that a_k = n, for all k ∈ ℕ.(a) Construct an injection i: ℕ → Seq[ℕ].(b) Prove that there is no surjection f: ℕ → Seq[ℕ]. (Consider the arguments of the proof of Cantor’s Theorem that we saw in class.)(c) Deduce from above that ℵ₀ < |Seq[ℕ]|. (Actually, it can be proven that |Seq[ℕ]| = |ℝ|.)

Answered: Image /qna-images/answer/4ee20e78-9fb0-4f9e-8a35-85fca84d26ae.jpg

Answered: = 7. Consider the set Seq[N] of all 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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You don't need part 1a, they are just saying it's a sequence.
8. Let fa,) be a sequence defined recursively by a, = 2 and for all n2 2, a,= a-1+ 2n. Prove via induction that a, = n? + n for all natural numbers n. 6
5. Compute the first five terms the sequence with nth term formula given by a =9n³ -10n². A) -1,32, 153, 416, 875 B) -1,1,-1,1,-1 C) −1, -2, -3, -4, – 5 D) 0,0,-4,-8,-16 E) -1, 32, 65, 98, 131 f. V % 5 t 6.0 g 6 b ABUS y h & 7 n U ★ 8 E ( 9 k O 0 р + 11 ? 0 1
3. What are the terms ao, a1, a2, and az of the sequence {an}, where an equals b) (n+ 1)"+l? d) [n/2] + [n/2]? a) 2" + 1? c) n/2]?
4. Prove the following statements using an inductive argument: _n(n+1)(2n+1) 6 72 a. Σiª: = i=1 12 b. If a = 2a- and b = 1-(1-a)², show that a = b for all natural numbers n. Assume that a is a fixed constant. c. Let a be the sequence defined by a₁ =1, a₁ = 8, and a₁ = a₁-11 +2a2 for n ≥ 3. Use an inductive argument to prove that if b₁ =3×2"-¹ +2(−1)", the a = b for all PL 72 neN
Find the indicated nth partial sum of (Sn) if the arithmetic sequence. a) -3 + -3/2 + 0 +…+30 b) u2= 8, u5= 9.5, n=12 c) u1=100, d=-5, n= 8
Let S be the set of all infinite sequences of “a”, “b”, or “c”. Thus S includes the string of “abc” repeated ad infinitum, “abcabcabcabc...”, as well as the string of “a” repeated, “aaaa...”, and the string with b’s at prime indicies and a’s elsewhere: “aabbababaaababaaa...”. Prove that the set S is larger than the set N. (Hint: recall that two infinite sets are the same size iff a bijection can be found to connect them in a 1-1 correspondence.)
Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers. For (i) take = xn= Show that {xn} is a Cauchy but not monotonic. For(ii), take= xn=n Show that{xn} is monotonic. Take &=;n E N; and n > N and m=n+1 Explain why (-1)" n |xn−Xn+1| > E Conclude that {xn) is monotonic but it is not Cauchy. For (iii), take xn = 1 +(-1)^. Show that {xn} is bounded, i.e. show that |xn| <2; for all n E N. Compute |xn-Xn+1| Explain why {xn} cannot be a Cauchy sequence. For (iv) Recall that every Cauchy sequence is bounded. Conclude that no such an example of a sequence {x} can be constructed.
Need help with this one please
c. Let a be the sequence defined by a₁ = 1, a₁ = 8, and a = a₁ +2a2 for n ≥ 3. Use an inductive argument to prove that if b = 3x2¹ +2(-1)", the a = b for all " neN
answer the following questions?
Find an explicit formula for a sequence of the form a an 1 + 1, 1 6 + 1 1 ㄹㅊ + 1 1 3' 8 11 221 231 1 1 1 1 4 9 호 10 + + 긍 with the initial terms given below.

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