Problem 3.Consider the following two sequences of real numbers4n – 31+1211+321Yn =2n22n2"In this exercise we will show that they are convergent.(a) Show that (x,) is eventually decreasing and bounded below. By eventuallydecreasing it is meant thatfor large enough n E N.(b) Show that (yn) is increasing and bounded above.Xn+1 < Xn,Observation: By the Monotone Convergence Limit, you have proven that the limit of(Yn) actually exists. It is a real challenge to show that it is actually n2/6.

(a) Given that xn=4n-32n Take n=1, x1=4-32=12=0.5 Take n=2, x2=8-322=54=1.25 So x2<x1 Assume…

4n – 3 1 1 + + 22 1 1 Yn = 12 ... 2n 32 n2 In this exercise we will show that they are convergent. (a) Show that (xn) is eventually decreasing and bounded below. By eventually decreasing it is meant that Xn+1 < Xn, for large enough n E N. (b) Show that (yn) is increasing and bounded above.

4n – 3 1 1 + + 22 1 1 Yn = 12 ... 2n 32 n2 In this exercise we will show that they are convergent. (a) Show that (xn) is eventually decreasing and bounded below. By eventually decreasing it is meant that Xn+1 < Xn, for large enough n E N. (b) Show that (yn) is increasing and bounded above.

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

Problem 3.
Consider the following two sequences of real numbers
4n – 3
1
+
12
1
1
+
32
1
Yn =
2n
22
n2"
In this exercise we will show that they are convergent.
(a) Show that (x,) is eventually decreasing and bounded below. By eventually
decreasing it is meant that
for large enough n E N.
(b) Show that (yn) is increasing and bounded above.
Xn+1 < Xn,
Observation: By the Monotone Convergence Limit, you have proven that the limit of
(Yn) actually exists. It is a real challenge to show that it is actually n2/6.

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