#1 27 Prove that lim 777-20 1²-20 216 # 3 #2 Define a =0 - (give Define a sequence {ant by selling a, = 1, Apri= √2an+1. Is {any convergent, either prove disprore. 24~ bet Fant be a sequence such that lim Anti-an=0. conclude that fant is convergent. Can you. an E-nc feet). de
#1 27 Prove that lim 777-20 1²-20 216 # 3 #2 Define a =0 - (give Define a sequence {ant by selling a, = 1, Apri= √2an+1. Is {any convergent, either prove disprore. 24~ bet Fant be a sequence such that lim Anti-an=0. conclude that fant is convergent. Can you. an E-nc feet). de
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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I need help solving those math problems.
![**Analysis: Homework Set 2**
**#1** Prove that \(\lim_{n \to \infty} \frac{n}{n^2 + 20} = 0\). (Provide an \(\epsilon\)-\(N\) proof.)
**#2** Define a sequence \(a_n\) by setting \(a_1 = 1\), \(a_{n+1} = \sqrt{2a_n + 1}\). Is \(a_n\) convergent? Either prove or disprove.
**#3** Let \(a_n\) be a sequence such that \(\lim_{n \to \infty} (a_{n+1} - a_n) = 0\). Can you conclude that \(a_n\) is convergent?
**#4** Define the sequence \(a_n\) as follows:
\[
a_n =
\begin{cases}
\frac{(2k+1)^k}{2k+3} & \text{if } n=3k \\
\frac{1}{k^k} & \text{if } n=3k+1 \\
(-1)^k & \text{if } n=3k+2
\end{cases}
\]
Find \(\limsup_{n \to \infty} a_n\) and \(\liminf_{n \to \infty} a_n\).
**#5** Discuss the convergence of the series:
\[
\sum_{n=1}^{\infty} \frac{(\ln n)^3}{\sqrt{n}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe742bcfc-6571-4eb6-8020-0d244ddd1492%2F5780329c-0710-4231-b605-17d2bd86342b%2Fcfu7bpl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Analysis: Homework Set 2**
**#1** Prove that \(\lim_{n \to \infty} \frac{n}{n^2 + 20} = 0\). (Provide an \(\epsilon\)-\(N\) proof.)
**#2** Define a sequence \(a_n\) by setting \(a_1 = 1\), \(a_{n+1} = \sqrt{2a_n + 1}\). Is \(a_n\) convergent? Either prove or disprove.
**#3** Let \(a_n\) be a sequence such that \(\lim_{n \to \infty} (a_{n+1} - a_n) = 0\). Can you conclude that \(a_n\) is convergent?
**#4** Define the sequence \(a_n\) as follows:
\[
a_n =
\begin{cases}
\frac{(2k+1)^k}{2k+3} & \text{if } n=3k \\
\frac{1}{k^k} & \text{if } n=3k+1 \\
(-1)^k & \text{if } n=3k+2
\end{cases}
\]
Find \(\limsup_{n \to \infty} a_n\) and \(\liminf_{n \to \infty} a_n\).
**#5** Discuss the convergence of the series:
\[
\sum_{n=1}^{\infty} \frac{(\ln n)^3}{\sqrt{n}}
\]
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