Problem 3. (10+10 points) Consider the following two sequences of real numbers 4n – 3 1 1 Yn = 12 1 + 22 1 + n2 Xn = 2n 32 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
Problem 3. (10+10 points) Consider the following two sequences of real numbers 4n – 3 1 1 Yn = 12 1 + 22 1 + n2 Xn = 2n 32 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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Problem 3.** (10+10 points) Consider the following two sequences of real numbers
\[ x_n = \frac{4n - 3}{2^n}, \,\, y_n = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}. \]
In this exercise we will show that they are convergent.
(a) Show that \((x_n)\) is eventually decreasing and bounded below. By eventually decreasing it is meant that
\[ x_{n+1} \leq x_n, \]
for large enough \( n \in \mathbb{N} \).
(b) Show that \((y_n)\) is increasing and bounded above.
**Observation:** By the Monotone Convergence Limit, you have proven that the limit of \((y_n)\) actually exists. It is a real challenge to show that it is actually \(\pi^2/6\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb791f969-1b7a-4924-ab3a-a621fcc29e59%2F12738187-232a-44a1-8d36-7c903e6b2769%2Feh55iyx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3.** (10+10 points) Consider the following two sequences of real numbers
\[ x_n = \frac{4n - 3}{2^n}, \,\, y_n = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}. \]
In this exercise we will show that they are convergent.
(a) Show that \((x_n)\) is eventually decreasing and bounded below. By eventually decreasing it is meant that
\[ x_{n+1} \leq x_n, \]
for large enough \( n \in \mathbb{N} \).
(b) Show that \((y_n)\) is increasing and bounded above.
**Observation:** By the Monotone Convergence Limit, you have proven that the limit of \((y_n)\) actually exists. It is a real challenge to show that it is actually \(\pi^2/6\).
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