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
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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.
Transcribed Image Text: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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