Looking for some help to prove or disprove #5. 2n + 11. Use the definition to prove that the sequenceis CauchynEN2. Let x E (0, 1), use e – d to proveV4+x – 2lim143. Let a, b, c, d E R. Define explicitly a bijection from [a, b] onto [c, d].an4. Let a1 = 1,An+1 =for n > 1. Show that lim an exists and find the limit.V 25. Prove or disprove that if {a,} and {b,} are Cauchy and b, # 0. ThenSan lis Cauchy

If an and bn are Cauchy and bn≠0, then anbn is not Cauchy. Result used: If an and bn are Cauchy and…

Answered: Prove or disprove that if {a,} and {b,}…

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Prove or disprove that if {a,} and {b,} are Cauchy and b,n # 0. Then An is Cauchy

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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Question

Looking for some help to prove or disprove #5.

2n + 1
1. Use the definition to prove that the sequence
is Cauchy
nEN
2. Let x E (0, 1), use e – d to prove
V4+x – 2
lim
1
4
3. Let a, b, c, d E R. Define explicitly a bijection from [a, b] onto [c, d].
an
4. Let a1 = 1,
An+1 =
for n > 1. Show that lim an exists and find the limit.
V 2
5. Prove or disprove that if {a,} and {b,} are Cauchy and b, # 0. Then
San l
is Cauchy

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kindly answer it perfectly
Correct and detailed answer
The remaining problems require you to construct a mathematical proof by induction. Remember, if you don't make use of the inductive hypothesis, there is no way to finish the proof correctly! 6. Prove the following inequality for all integers n 21: n! ≤ nn .
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Just need help with first two questions.
To prove the statement " 8 does not divide x² – 1 if x is even." directly. (choose the correct answer of the proof) Assume that Ox is odd. Ox is even. 08 does not divide x2 – 1 and x is odd. 08 divides x² 1 and x is even. 08 divides x? – 1. then Ox = 2k + 1. Ox = 2k for some integer k. Ox? – 1 = 8k. %3D 08 = k(x² – 1) So Ox? – 1 = 4k2 + 4k = 4k(k + 1). Since k(k + 1) is even then 8 divides x2 – 1. Hence, result follows. Ox? – 1 = 4k2 – 1 = 4k? – 2+ 1 = 2(2k² – 1) + 1 which is odd. So 8 does not divide x² – 1. Hence, result follows. Ox? = 8k + 1 = 2(4k) + 1. So x² is odd and so is x. Hence, result follows. 08 = k(x – 1)(x + 1) sincex – 1 and x + 1 have the same parity then if they are both odd and k is odd we get a contradiction. So - result follows.
Prove or disprove the following statement, (c):
Complete the conjecture based on the pattern you observe in the examples below. 5. Conjecture: The sum of any two odd numbers is 1+1=2 1+5=6 7+9=16 9 + 11 = 20 13 +21 = 34 101 + 103 = 204 6. Conjecture: The product of any two odd numbers is 1x1 = 1 1x3 = 3 3x7 = 21 1.2 Patterns and Conjectures 9x1199 13x11 = 143 101x103 = 10,403 7. Conjecture: The product of a number (x - 1) and the number (x + 1) is always equal to 3x5 = 4² - 1 2 7x9 = 8²-1 4x6 = 5² - 1 6x8 = 7² - 1 5x7 = 6²-1 8x10 = 9²-1 Page 1 of 1
Question 1. Prove that For all integers n. ·N·・閱 - = n 4
Number theory problem: part c
Which of the following are propositions (assume that x is a real number)? If not, explain why. (a) 2π+5π=7π. (b) The product of x2 and x3 is x6. (c) It is not possible for 315 − 7 to be both even and odd. (d) If the integer x is odd, is x2 odd? (e) The integer 2524287 − 1 is prime. (f) 1.7+.2=4.0.