Title: Math 2260 Chapter 9 (Sequences and Series)

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

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

Q: [latex]
$$\lim_{n \rightarrow \infty} \frac{\ln n}{n} = $$
[/latex]
A: [latex]
$$ \lim_{n \rightarrow \infty} \frac{\ln n}{n} = 0 $$
[/latex]

Q: [latex]
$$\lim_{n \rightarrow \infty} n^{1/n} =$$
[/latex]
A: [latex]
$$\lim_{n \rightarrow \infty} n^{1/n} =  1 $$
[/latex]

Q: [latex]
$$\lim_{n \rightarrow \infty} x^{1/n} = $$
[/latex]
A: [latex]
$$\lim_{n \rightarrow \infty} x^{1/n} =  1 \quad \text{ if } x > 0$$
[/latex]

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

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

Q: [latex]
$$\lim_{n \rightarrow \infty} \frac{x^n}{n!} =   $$
[/latex]
A: [$$]
\lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0 \text{ for any } x
[/$$]

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

Q: [latex]The geometric series
$$\sum_{n=1}^\infty a r^{n-1}$$
converges for what values of $a$ and $r$ (and to what value)?
[/latex]
A: [latex]
For any $a$, the geometric series
$$\sum_{n=1}^\infty a r^{n-1} = \frac{a}{1 - r}, \text{ if } |r| < 1.$$
and diverges for all other $r$.
[/latex]

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

Q: [latex]The $n$-th term test. \\
If $\lim_{n \rightarrow \infty} a_n = 0$, what do we know about the convergence of the series $\sum a_n$?
[/latex]
A: [latex]
Nothing. The series might diverge (like the harmonic series $\sum 1/n$) or it might converge (like the $p$-series $\sum 1/n^2$).
[/latex]

Q: [latex]The integral test for series. \\
If $a_n = f(n)$ where $f$ is a continuous, positive, decreasing function, then
$$\sum_{n=1}^{\infty} a_n$$
is bounded above and below by:
[/latex]
A: [latex]
$$\sum_{n=1}^k a_n + \int_{k+1}^\infty f(x) \, dx \leq \sum_{n=1}^{\infty} a_n \leq \sum_{n=1}^{k} a_n + \int_k^\infty f(x) \, dx.$$
Note: If $a_n = f(n)$and $f(x)$ is not continuous, positive and decreasing for all $x$, but is continuous, positive, and decreasing for large enough $x$, you are supposed to split off the first (finite) part of the sum and reindex, then apply this test to the new series. If the new series converges (or diverges), the old one does, too.
[/latex]

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

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

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

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

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

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

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

Q: [latex]The Alternating Series Test. \\
The series
$$\sum_{n=1}^{\infty} (-1)^n u_n =$$
converges if
[/latex]
A: [latex]
The $u_n$ are positive, decreasing, and $\lim_{n \rightarrow \infty} u_n = 0$.
[/latex]

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

Q: [latex]
The ``$n$-th degree Taylor polynomial" or ``Taylor polynomial of order $n$" for $f(x)$ centered at $x=a$ is
[/latex]
A: [latex]
$$P_{n}(x) = f(a) + f'(a) (x-a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n. $$
[/latex]

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

Q: [latex]
If the Taylor series
$$f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} (x-a)^k $$
centered at $a$ converges at $x$, does it converge to $f(x)$? [/latex]
A: [latex]
Not always! Taylor's Theorem guarantees this if \\
$f(x)$ has derivatives of all orders on an open interval including $x$ and $a$.\\
For each $k$, there is a bound
$$M_k > |f^{(k)}(y)|$$
for $y$ between $a$ and $x$ so that
$$\lim_{k \rightarrow \infty} \frac{M_k}{k!} (x-a)^k \rightarrow 0$$
[/latex]

Q: [latex]The Taylor series for $e^x$ \\
(centered at $x=0$) is[/latex]
A: [latex]
$$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + \cdots$$
for all $x$.
[/latex]

Q: [latex]The Taylor series for $\sin x$ \\
(centered at $x = 0$) is [/latex]
A: [latex]
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots + (-1)^k \frac{x^{2k+1}}{(2k+1)!} + \cdots$$
for all $x$.
[/latex]

Q: [latex]The Taylor series for $\cos x$\\
 (centered at $x = 0$) is[/latex]
A: [latex]
$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots + (-1)^k \frac{x^{2k}}{(2k)!} + \cdots$$
for all $x$.
[/latex]

Q: [latex]The Taylor Series for $f(x)$ centered at $x = a$ is [/latex]
A: [latex]
$$f(x) = f(a) + f'(a) (x - a) + \frac{f''(a)}{2!} (x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x - a)^n + \cdots $$
where $f^{(n)}(a)$ is the $n$-th derivative of $f(x)$, evaluated at $x = a$ (a number, not a function).
[/latex]

Q: [latex]The Taylor series for $\frac{1}{1-x}$\\
 (centered at $x = 0$) is[/latex]
A: [latex]
$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \dots + x^n + \dots $$
when $x \in (-1,1)$.
[/latex]

Q: [latex]The Taylor series for $\frac{1}{1+x}$ \\
(centered at $x = 0$) is[/latex]
A: [latex]
$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - \cdots + (-x)^n + \cdots $$
when $x \in (-1,1)$.
[/latex]

Q: [latex]The Taylor series for $\operatorname{arctan} x$\\
 (centered at $x = 0$) is[/latex]
A: [latex]
$$\operatorname{arctan} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots + (-1)^n \frac{x^{2n+1}}{2n+1} + \dots$$
for $x \in [-1,1]$
[/latex]

Q: [latex]The Taylor series for $\ln (1+x)$\\
 (centered at $x = 0$) is [/latex]
A: [latex]
$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots + (-1)^n \frac{x^{n}}{n} + \dots$$
for $-1 < x \leq 1$.
[/latex]