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)

*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)

*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)

*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)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be True.

For example, choice (d) should be False.

*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!*

*False**True**True**False*

${\sum}_{n=0}^{\infty}{\left(-x\right)}^{n}$ is
the Taylor series for which function? Exactly one option must be correct)

*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)

*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)

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be True.

For example, choice (d) should be True.

For example, choice (e) should be False.

*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!*

*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.*True*This geometric series converges to $\frac{1}{1-0.3}=\frac{10}{7}.$*True*This series converges to $e.$*True*This series is just the series for $sinx$ evaluated at the point $x=\pi $. It converges to $sin\pi =0$.*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)

*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 $a\u2215b$,
where $a$ and
$b$ are integers. What is
the numerator when $a\u2215b$
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

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

*Please try again.*

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

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)

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*

*Choice (e) is incorrect*

*Choice (f) is incorrect*