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

Name: **Problem 3.** (10+10 points) Consider the following two sequences of
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Description: **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 convergen

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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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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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 $\pi^2/6$.

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