Fundamentals of Mathematical Analysis

Rod Haggarty

Chapter 3 Sequences - all with Video Answers

Educators

Section 1

Convergent sequences

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Problem 1

Use the definition of a convergent sequence to prove the following:
(a) $\frac{n-1}{2 n} \rightarrow \frac{1}{2}$ as $n \rightarrow \infty$
(b) $\frac{(-1)^{n}}{n^{2}} \rightarrow 0$ as $n \rightarrow \infty$

Nick Johnson

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01:25

Problem 2

Use the rules for convergent scquences to establish that if $\left(a_{n}\right)$ converges to $A$ and $\left(b_{n}\right)$ converges to $B$ then $\left(\alpha a_{n}+\beta b_{n}\right)$ converges to $\alpha A+\beta B$

Wendi Zhao

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00:33

Problem 3

Use the rules for convergent scquences to evaluate the following limits:
(a) $\lim _{n \rightarrow \infty} \frac{4 n^{3}+6 n-7}{n^{3}-2 n^{2}+1}$
(b) $\lim _{n \rightarrow \infty} \frac{6-n^{2}}{n^{2}+5 n}$
(c) $\lim _{n \rightarrow \infty}\left[\log _{e}(n+1)-\log _{e} n\right]$

Nick Johnson

Numerade Educator

01:01

Problem 4

Use the sandwich rule to prove that each of the following sequences $\left(a_{n}\right)$ converges to zero:
(a) $a_{n}=\frac{(-1)^{n} n}{\sqrt{n^{3}}+1}$
(b) $a_{n}=\cos n$

Tyler Moulton

Numerade Educator