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Magazine Problem 56
Compute \(N_{t}\) and \(N_{t} / N_{t-1}\) for \(t=2,3,4, \ldots .20\) when $$ N_{t+1}=N_{t}+2 N_{t-1} $$ with \(N_{0}=1\) and \(N_{1}=1\).
Problem 56
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{n^{2}}, \epsilon=0.001 $$
Problem 57
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.1 $$
Problem 58
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.05 $$
Problem 58
$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{7} N_{t} \text { with } N_{0}=6400 $$
Problem 59
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{(-1)^{n}}{n}, \epsilon=0.01 $$
Problem 60
In Problems , graph the line \(N_{t+1}=R N_{t}\) in the \(N_{t}-N_{t+1}\) plane for the indicated value of \(R\) and locate the points \(\left(N_{t}, N_{t+1}\right), t=0\), 1\. and 2. for the given value of \(N_{0}\). $$ R=2, N_{0}=3 $$
Problem 60
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{(-1)^{n}}{n}, \epsilon=.001 $$
Problem 61
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{n}{n+1}, \epsilon=0.01 $$
Problem 62
Formal Definition of Limits: \lim _{n \rightarrow \infty} a_{n}=a .\( Find the limit \)a\(, and determine \)N\( so that \)\left|a_{n}-a\right|<\epsilon\( for all \)n>N\( for the given value of \)\epsilon$. $$ a_{n}=\frac{n+1}{n}, \epsilon=.05 $$
