MATH1001 Quizzes

Quiz 12: Taylor series
Question 1 Questions
Find the coefficient of ${x}^{2}$ in the Taylor series about $x=0$ for $f\left(x\right)={e}^{-{x}^{2}}$. (Hint: start with the Taylor series for ${e}^{x}$.) Exactly one option must be correct)
 a) $1∕4$ b) $-1$ c) $1∕2$ d) $-2$ e) $1$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the coefficient of ${x}^{3}$ in the Taylor series about $x=0$ for $f\left(x\right)=sin2x$. (Hint: start with the Taylor series for $sinx$.) Exactly one option must be correct)
 a) $-2∕3$ b) $-4∕3$ c) $4∕3$ d) $-8∕3$ e) $2∕3$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which one of the following is the Taylor series for $sinx$? Exactly one option must be correct)
 a) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n-1}}{\left(2n-1\right)!}$ b) $\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n+1}{x}^{2n+1}}{\left(2n+1\right)!}$ c) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n+1}}{\left(2n+1\right)!}$ d) $\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n+1}}{\left(2n+1\right)!}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Which of the following are true statements? (Zero or more options can be correct)
 a) Every function $f\left(x\right)$ is equal to its Taylor series for all real $x$. b) There exist functions $f\left(x\right)$ which are equal to their Taylor series for all real $x$. c) There exist functions $f\left(x\right)$ which are equal to their Taylor series for some, but not all, real numbers $x$. d) A function $f\left(x\right)$ can never equal its Taylor series. The Taylor series is only ever an approximation to the function.

There is at least one mistake.
For example, choice (a) should be False.
There is at least one mistake.
For example, choice (b) should be True.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be False.
Correct!
1. False
2. True
3. True
4. False
${\sum }_{n=0}^{\infty }{\left(-x\right)}^{n}$ is the Taylor series for which function? Exactly one option must be correct)
 a) $sinx$ b) $cosx$ c) $\frac{1}{1-x}$ d) $\frac{1}{1+x}$ e) ${e}^{x}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the Taylor series about $x=0$ for $f\left(x\right)=\frac{1}{{\left(1-x\right)}^{2}}$. Exactly one option must be correct)
 a) $\sum _{n=0}^{\infty }\left(n+1\right){x}^{n}$ b) $\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\left(n+1\right){x}^{n}$ c) $\sum _{n=0}^{\infty }n{x}^{n}$ d) $\sum _{n=0}^{\infty }{x}^{2n}$ e) $\sum _{n=0}^{\infty }-{x}^{2n}$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Several of the following series of numbers converge. Check all those that do. (Zero or more options can be correct)
 a) $1+2+3+4+5+\dots \dots$ b) $1+0.3+0.{3}^{2}+0.{3}^{3}+0.{3}^{4}+\dots \dots$ c) $1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\dots \dots$ d) $\pi -\frac{{\pi }^{3}}{3!}+\frac{{\pi }^{5}}{5!}-\frac{{\pi }^{7}}{7!}+\dots \dots$ e) $1-2+3-4+5-6+\dots \dots$

There is at least one mistake.
For example, choice (a) should be False.
Adding up the first $n$ terms of this series gives $1+2+3+\dots \dots +n=\frac{n\left(n+1\right)}{2}$ As $n$ becomes arbitrarily large, this sum approaches infinity. Hence the series diverges.
There is at least one mistake.
For example, choice (b) should be True.
This geometric series converges to $\frac{1}{1-0.3}=\frac{10}{7}.$
There is at least one mistake.
For example, choice (c) should be True.
This series converges to $e.$
There is at least one mistake.
For example, choice (d) should be True.
This series is just the series for $sinx$ evaluated at the point $x=\pi$. It converges to $sin\pi =0$.
There is at least one mistake.
For example, choice (e) should be False.
The partial sums are ${S}_{1}=1,\phantom{\rule{0.3em}{0ex}}{S}_{2}=-1,\phantom{\rule{0.3em}{0ex}}{S}_{3}=2,\phantom{\rule{0.3em}{0ex}}{S}_{4}=-2,\phantom{\rule{0.3em}{0ex}}{S}_{5}=3,\phantom{\rule{0.3em}{0ex}}{S}_{6}=-3$ and so on. We see that adding up the first $2n$ terms gives $-n$ while adding up the first $2n-1$ terms gives $n$. Clearly these partial sums never settle down to a limit as $n$ approaches infinity, so the series diverges.
Correct!
1. False Adding up the first $n$ terms of this series gives $1+2+3+\dots \dots +n=\frac{n\left(n+1\right)}{2}$ As $n$ becomes arbitrarily large, this sum approaches infinity. Hence the series diverges.
2. True This geometric series converges to $\frac{1}{1-0.3}=\frac{10}{7}.$
3. True This series converges to $e.$
4. True This series is just the series for $sinx$ evaluated at the point $x=\pi$. It converges to $sin\pi =0$.
5. False The partial sums are ${S}_{1}=1,\phantom{\rule{0.3em}{0ex}}{S}_{2}=-1,\phantom{\rule{0.3em}{0ex}}{S}_{3}=2,\phantom{\rule{0.3em}{0ex}}{S}_{4}=-2,\phantom{\rule{0.3em}{0ex}}{S}_{5}=3,\phantom{\rule{0.3em}{0ex}}{S}_{6}=-3$ and so on. We see that adding up the first $2n$ terms gives $-n$ while adding up the first $2n-1$ terms gives $n$. Clearly these partial sums never settle down to a limit as $n$ approaches infinity, so the series diverges.
What is the value of $0.77777777\dots \dots$ ? Exactly one option must be correct)
 a) $\frac{7}{10}$ b) $\frac{7}{9}$ c) $\frac{70}{91}$ d) $\frac{69}{90}$ e) None of the above

Choice (a) is incorrect
Choice (b) is correct!
Note that $0.77777777\dots \dots =\frac{7}{10}+\frac{7}{100}+\frac{7}{1000}+\dots \dots =\frac{7}{10}\left(1+\frac{1}{10}+\frac{1}{100}+\dots \dots \right)$ The geometric series $1+\frac{1}{10}+\frac{1}{100}+\dots \dots$ converges to $\frac{10}{9}.$ Hence $0.77777777\dots \dots =\frac{7}{9}.$
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
The value of $0.5454545454\dots \dots$ is a rational number $a∕b$, where $a$ and $b$ are integers. What is the numerator when $a∕b$ is expressed as a fraction in its lowest form? Enter your answer into the box.

Correct!
$0.5454545454\dots \dots =\left(\frac{5}{10}+\frac{5}{1{0}^{3}}+\frac{5}{1{0}^{5}}+\dots \dots \right)+\left(\frac{4}{1{0}^{2}}+\frac{4}{1{0}^{4}}+\frac{4}{1{0}^{6}}+\dots \dots \right)$

Therefore since

$\frac{5}{10}+\frac{5}{1{0}^{3}}+\frac{5}{1{0}^{5}}+\dots \dots =\frac{50}{99}$

and

$\frac{4}{1{0}^{2}}+\frac{4}{1{0}^{4}}+\frac{4}{1{0}^{6}}+\dots \dots =\frac{4}{99},$

we see that

$0.5454545454\dots \dots =\frac{50}{99}+\frac{4}{99}=\frac{54}{99}=\frac{6}{11}.$

$0.5454545454\dots \dots =\left(\frac{5}{10}+\frac{5}{1{0}^{3}}+\frac{5}{1{0}^{5}}+\dots \dots \right)+\left(\frac{4}{1{0}^{2}}+\frac{4}{1{0}^{4}}+\frac{4}{1{0}^{6}}+\dots \dots \right)$

Now find the sum of each of these two separate geometric series.

Which series equals the Taylor series of $cosh\left(-{x}^{2}\right)$ ? Exactly one option must be correct)
 a) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{4n}}{\left(2n\right)!}$ b) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n-1}}{\left(2n-1\right)!}$ c) $\sum _{n=0}^{\infty }\frac{{x}^{4n}}{\left(2n\right)!}$ d) $\sum _{n=1}^{\infty }\frac{{x}^{4n}}{\left(2n\right)!}$ e) $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}{x}^{2n}}{\left(2n\right)!}$ f) None of the above.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect
Choice (f) is incorrect