Chapter 2

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.1 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{1}{\sqrt{n}}, \epsilon=0.05 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\frac{(-1)^{n}}{n}, \epsilon=0.01 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=e^{-3 n}, \epsilon=0.001 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\ln \left(1+\frac{1}{n}\right), \epsilon=0.1 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=2^{-n}, \epsilon=0.01 $$

\(\lim _{n \rightarrow \infty} a_{n}=a\). Find the limit \(a\), and determine \(N\) so that \(\left|a_{n}-a\right|<\epsilon\) for all \(n>N\) for the given value of \(\epsilon\) $$ a_{n}=\log \left(1+\frac{2}{n^{2}}\right), \epsilon=0.05 $$

In Problems \(73-78\), use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{3}{n}=0 $$

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n+1}=0 $$

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0 $$

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} e^{-2 n}=0 $$

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} 2^{-3 n}=0 $$

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{n}{n+1}=1 $$

In Problems 79-90, use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{2}{n^{2}}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{m \rightarrow \infty}\left(\frac{2}{n}-\frac{3}{n^{2}+1}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{2 n-3}{\sqrt{n}}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n^{2}+1}{n^{2}}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-5}{n}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n+1}{n^{2}-1}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(\frac{n^{2}}{n^{2}+4}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty}\left(3^{-n}-4^{-n}\right) $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+2^{-n}}{n} $$

Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{1+e^{-n}}{n} $$

In Problems 91-100, the sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}, a_{0}=1 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}, a_{0}=3 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=-2 a_{n}, a_{0}=1 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=-2 a_{n}, a_{0}=2 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=1+2 a_{n}, a_{0}=0 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=4-2 a_{n}, a_{0}=\frac{4}{3} $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=\frac{a_{n}}{1+a_{n}}, a_{0}=1 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=\sqrt{a_{n}}, a_{0}=16 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=a_{n}+\frac{1}{a_{n}}, a_{0}=1 $$

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}^{2}, a_{0}=1 $$

In Problems 101-110, the sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{1}{2} a_{n}+2 $$

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{1}{3} a_{n}+\frac{4}{3} $$

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{5}{2}-\frac{1}{2} a_{n} $$

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=a_{n}^{2}-a_{n} $$

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\sqrt{5 a_{n}} $$

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\sqrt{a_{n}+2} $$

In Problems 111-118, assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{3} a_{n}+\frac{4}{3}, a_{0}=0 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\sqrt{2 a_{n}}, a_{0}=1 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{3}{a_{n}+2}, a_{0}=0 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=2 a_{n}\left(1-a_{n}\right), a_{0}=0.1 $$

Assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed points of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which fixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{9}{a_{n}}\right), a_{0}=-1 $$

In Problems 119-124, write each sum in expanded form. $$ \sum_{k=1}^{4} \sqrt{k} $$

Write each sum in expanded form. $$ \sum_{k=3}^{5}(k-1)^{2} $$

Write each sum in expanded form. $$ \sum_{k=2}^{6} 3^{k} $$

Write each sum in expanded form. $$ \sum_{k=1}^{3} \frac{k^{2}}{k^{2}+1} $$

Write each sum in expanded form. $$ \sum_{n=0}^{3} a_{n} \text { where } a_{0}=1 \text { and } a_{n+1}=2 a_{n} $$

Write each sum in expanded form. $$ \sum_{n=0}^{3} a_{n} \text { where } a_{0}=1 \text { and } a_{n+1}=2 a_{n} $$