Title: Math 2260 Chapter 9 (Sequences and Series)

Q: [latex]
\begin{large}
The sequence $\{ a_n \}$ converges to $L$ if
\end{large}
[/latex]
A: [latex]
\begin{large}
for every $\epsilon > 0$ there exists some $N$ so that
\begin{equation*}
|a_n - L| < \epsilon
\end{equation*}
whenever $n \geq N$.
\end{large}
[/latex]

Q: [latex]
\begin{large}
Suppose $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are sequences so that
\begin{itemize}
\item $a_n \leq b_n \leq c_n$
\item $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} c_n = L$
\end{itemize}
then
\end{large}
[/latex]
A: [latex]
\begin{large}
then $\lim_{n \rightarrow \infty} b_n = L$ as well.
\end{large}
[/latex]

Q: [latex]
\begin{huge}
\begin{equation*}\lim_{n \rightarrow \infty} \frac{\ln n}{n} = \end{equation*}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*} \lim_{n \rightarrow \infty} \frac{\ln n}{n} = 0 \end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
 \begin{equation*} \lim_{n \rightarrow \infty} n^{1/n} =  \end{equation*}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*} \lim_{n \rightarrow \infty} n^{1/n} =  1 \end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} x^{1/n} =
\end{equation*}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} x^{1/n} =  1 \quad \text{ if } x > 0
\end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} x^n =
\end{equation*}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} x^n = 0, \quad \text{ if } |x| < 1
\end{equation*}
and does not exist if $|x| > 1$.
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n}\right)^n =
\end{equation*}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n}\right)^n =  e^x \quad \text{ for any } x
\end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} \frac{x^n}{n!} =
\end{equation*}
\end{huge}
[/latex]
A: [latex]
\begin{huge}
\begin{equation*}
\lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0 \text{ for any } x
\end{equation*}
\end{huge}
[/latex]

Q: [latex]
\begin{large}
Define the $k$-th \emph{partial sum} $s_k$ of a series
\begin{equation*}
\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots
\end{equation*}
\end{large}
[/latex]
A: [latex]
\begin{large}
The $k$-th \emph{partial sum} $s_n$ of a series is
\begin{equation*}
s_k = \sum_{n=1}^k a_n = a_1 + \cdots + a_k.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The geometric series
\begin{equation*}
\sum_{n=1}^\infty a r^{n-1}
\end{equation*}
converges when (and to what value)?
\end{large}
[/latex]
A: [latex]
\begin{large}
The geometric series
\begin{equation*}
\sum_{n=1}^\infty a r^{n-1} = \frac{a}{1 - r}, \text{ if } |r| < 1.
\end{equation*}
and diverges for all other $r$.
\end{large}
[/latex]

Q: [latex]
\begin{huge}
The $n$-th term test. \\
If $\lim_{n \rightarrow \infty} a_n$ \textbf{does not exist} or \textbf{is not zero}, what do we know about the convergence of $\sum a_n$?
\end{huge}
[/latex]
A: [latex]
\begin{huge}
The $n$-th term test. \\
If $\lim_{n \rightarrow \infty} a_n$ does not exist or is not zero, then the series $\sum a_n$ \textbf{diverges}.
\end{huge}
[/latex]

Q: [latex]
\begin{huge}
The $n$-th term test. \\
If $\lim_{n \rightarrow \infty} a_n = 0$, what do we know about the convergence of the series $\sum a_n$?
\end{huge}
[/latex]
A: [latex]
\begin{huge}
The $n$-th term test. \\
If $\lim_{n \rightarrow \infty} a_n = 0$, then the series $\sum a_n$ \textbf{may} or \textbf{may not} converge. \\
Think ``harmonic series ($\sum 1/n$) diverges''.
\end{huge}
[/latex]

Q: [latex]
\begin{large}
The integral test for series. \\
If $a_n = f(n)$ where $f$ is a continuous, positive, decreasing function, then
\begin{equation*}
\sum_{n=1}^{\infty} a_n
\end{equation*}
is bounded above and below by:
\end{large}
[/latex]
A: [latex]
\begin{large}
If $a_n = f(n)$ where $f$ is a continuous, positive, decreasing function, then
\begin{equation*}
\int_{n+1}^\infty f(x) \, dx \leq \sum_{n=1}^{\infty} a_n \leq \int_n^\infty f(x) \, dx.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
The $p$-series test.\\
When does
\begin{equation*} \sum_{n=1}^\infty \frac{1}{n^p}  \end{equation*}
converge?
[/latex]
A: [latex]
The $p$-series test.\\
The series
\begin{equation*} \sum_{n=1}^\infty \frac{1}{n^p}  \end{equation*} conveges if and only if $p > 1$.
[/latex]

Q: [latex]
\begin{large}
The Comparison Test (for Series).\\
Given series $\sum a_n$, $\sum b_n$, $\sum c_n$ with non-negative terms and
\begin{equation*}a_n \leq b_n \leq c_n\end{equation*}
what do we know about the convergence (or divergence) of the three series?
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{enumerate}
\item If $\sum c_n$ converges, $\sum b_n$ converges.
\item If $\sum a_n$ diverges, $\sum b_n$ diverges.
\end{enumerate}
\end{large}
[/latex]

Q: [latex]
\begin{large}
Limit Comparison Test (for Series).\\
If $a_n > 0$ and $b_n > 0$ (for all $n$), and
\begin{equation*}
\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L
\end{equation*}
for $0 < L < \infty$, what do we know about the convergence of $\sum a_n$ and $\sum b_n$?
\end{large}
[/latex]
A: [latex]
\begin{center}
\textbf{Both} $\sum a_n$ and $\sum b_n$ \textbf{converge} \\
or \\
\textbf{both} $\sum a_n$ and $\sum b_n$ \textbf{diverge}.
\end{center}
[/latex]

Q: [latex]
\begin{large}
Limit Comparison Test (for Series).\\
If $a_n > 0$ and $b_n > 0$ (for all $n$), and
\begin{equation*}
\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 0,
\end{equation*}
what do we know about the convergence of $\sum a_n$ and $\sum b_n$?
\end{large}
[/latex]
A: [latex]
\begin{large}
If $\sum b_n$ converges, then $\sum a_n$ converges.
\end{large}
[/latex]

Q: [latex]
\begin{large}
Limit Comparison Test (for Series).\\
If $a_n > 0$ and $b_n > 0$ (for all $n$), and
\begin{equation*}
\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \infty
\end{equation*}
what do we know about the convergence of $\sum a_n$ and $\sum b_n$?
\end{large}
[/latex]
A: [latex]
\begin{large}
If $\sum b_n$ diverges, then $\sum a_n$ diverges.
\end{large}
[/latex]

Q: [latex]
\begin{large}
The Ratio Test (for Series). \\
If $a_n > 0$, and
\begin{equation*}
\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = p
\end{equation*}
what do we know about the convergence of $\sum a_n$?
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{enumerate}
\item Series \textbf{converges} if $p < 1$.
\item Series \textbf{diverges} if $p > 1$.
\item Test \textbf{is inconclusive} if $p = 1$.
\end{enumerate}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The Root Test (for Series). \\
If $a_n > 0$, and
\begin{equation*}
\lim_{n \rightarrow \infty} \sqrt[n]{a_n} = p
\end{equation*}
what do we know about the convergence of $\sum a_n$?
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{enumerate}
\item Series \textbf{converges} if $p < 1$.
\item Series \textbf{diverges} if $p > 1$.
\item Test \textbf{is inconclusive} if $p = 1$.
\end{enumerate}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The Alternating Series Test. \\
The series
\begin{equation*}
\sum_{n=1}^{\infty} (-1)^n u_n =
\end{equation*}
converges if
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{enumerate}
\item The $u_n$ are positive.
\item The $u_n$ are decreasing.
\item $\lim_{n \rightarrow \infty} u_n = 0$.
\end{enumerate}
\end{large}
[/latex]

Q: [latex]
The Taylor series for $f(x)$ centered at $a = 0$ is also called
[/latex]
A: [latex]
The \textbf{Maclaurin series}:
\begin{equation*}
f(0) + f'(0) x + \frac{f''(0)}{2} x^2 + \cdots + \frac{f^{(n)}(0)}{n!} x^n  + \cdots
\end{equation*}
[/latex]

Q: [latex]
\begin{large}
The $n$-th degree \textbf{Taylor polynomial} or \textbf{Taylor polynomial of order $n$} for $f(x)$ centered at $x=a$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
P_n(x) = f(a) + f'(a) (x-a) + \frac{f''(a)}{2} (x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
Taylor's Formula\\
If $f(x)$ has derivatives of all orders in an interval $I$ containing $a$, then for $x$ in $I$,
\begin{equation*} f(x) = f(a) + f'(a) (x-a) + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n + R_n(x).
\end{equation*}
What is the remainder term: $R_n(x)$?
[/latex]
A: [latex]
The remainder term is
\begin{equation*} R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}. \end{equation*}
where $c$ is some number in $[a,x]$.
[/latex]

Q: [latex]
If the Taylor series for $f(x)$ centered at $a$ converges at $x$, does it converge to $f(x)$?
[/latex]
A: [latex]
Not always! Guaranteed if
\begin{enumerate}
\item $f(x)$ has derivatives of all orders on an interval including $x$ and $a$.
\item There is a bound $M > |f^{(n)}(y)|$ for $y$ between $x$ and $x$.
\end{enumerate}
[/latex]

Q: [latex]
The Taylor series for $e^x$ is
[/latex]
A: [latex]
\begin{equation*}e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} +
\end{equation*}
and the series always converges to $e^x$.
[/latex]

Q: [latex]
The Taylor series for $\sin x$ is
[/latex]
A: [latex]
\begin{equation*}\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots + (-1)^k \frac{x^{2k+1}}{(2k+1)!} +
\end{equation*}
and the series always converges to $\sin x$.
[/latex]

Q: [latex]
The Taylor series for $\cos x$ is
[/latex]
A: [latex]
\begin{equation*}\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{4!} + \cdots + (-1)^k \frac{x^{2k}}{(2k)!} +
\end{equation*}
and the series always converges to $\cos x$.
[/latex]

Q: [latex]
The Taylor Series for $f(x)$ centered at $a$ is
[/latex]
A: [latex]
\begin{equation*} f(a) + f'(a) (x - a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x - a)^n + \cdots \end{equation*}
[/latex]

Q: [latex]
\begin{large}
The Taylor series for
\begin{equation*} \frac{1}{1-x} \end{equation*}
is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \dots + x^n + \dots \end{equation*}
when $|x| < 1$.
\end{large}
[/latex]

Q: [latex]
\begin{large}
The Taylor series for
\begin{equation*}\frac{1}{1+x}\end{equation*}
is
\end{large}
[/latex]
A: [latex]
\begin{large} \begin{equation*} \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - \cdots + (-x)^n + \cdots \end{equation*}
when $|x| < 1$.
\end{large}
[/latex]

Q: [latex]
\begin{large}
The Taylor series for $\operatorname{arctan} x $
is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \operatorname{arctan} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots + (-1)^n \frac{x^{2n+1}}{2n+1} + \dots
\end{equation*}
for $|x| \leq 1$.
\end{large}
[/latex]

Q: [latex]
\begin{large}
The Taylor series for $\ln (1+x) $
is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^n \frac{x^{n}}{n} + \dots
\end{equation*}
for $-1 < x \leq 1$.
\end{large}
[/latex]