#1 Prove that lim 216 27 1²20 =0 - (give an E-nc fecool).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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I need help solving analytical math problems.
![### Analysis Homework Set 2
#### Problem 1
Prove that
\[
\lim_{{n \to \infty}} \frac{n}{n^2 + 20} = 0
\]
(give an ε-n₀ proof).
#### Problem 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.
#### Problem 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?
#### Problem 4
Define \( a_n \) as follows:
\[
a_n =
\begin{cases}
\left( \frac{2k+1}{2k+3} \right)^k & \text{if } n = 3k \\
\frac{1}{k^k} & \text{if } n = 3k+1 \\
(-1)^k & \text{if } n = 3k+2
\end{cases}
\]
Find
\[
\limsup_{{n \to \infty}} a_n \quad \text{and} \quad \liminf_{{n \to \infty}} a_n
\]
#### Problem 5
Discuss the convergence of the series
\[
\sum_{{n=1}}^{\infty} \left( \frac{\ln n}{\sqrt{n}} \right)^3
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Transcribed Image Text:### Analysis Homework Set 2
#### Problem 1
Prove that
\[
\lim_{{n \to \infty}} \frac{n}{n^2 + 20} = 0
\]
(give an ε-n₀ proof).
#### Problem 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.
#### Problem 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?
#### Problem 4
Define \( a_n \) as follows:
\[
a_n =
\begin{cases}
\left( \frac{2k+1}{2k+3} \right)^k & \text{if } n = 3k \\
\frac{1}{k^k} & \text{if } n = 3k+1 \\
(-1)^k & \text{if } n = 3k+2
\end{cases}
\]
Find
\[
\limsup_{{n \to \infty}} a_n \quad \text{and} \quad \liminf_{{n \to \infty}} a_n
\]
#### Problem 5
Discuss the convergence of the series
\[
\sum_{{n=1}}^{\infty} \left( \frac{\ln n}{\sqrt{n}} \right)^3
\]
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