If you are reading this message either your browser does not support JavaScript or else JavaScript is not enabled. You will need to enable JavaScript and then reload this page before you can use this quiz.

Quiz 12: Taylor series

Question

Question 1

Find the coefficient of

x^{2}

in the Taylor series about

x = 0

for

f (x) = e^{- x^{2}}

. (Hint: start with the Taylor series for

e^{x}

Not correct. Choice (a) is false.

Your answer is correct.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 2

Find the coefficient of

x^{3}

in the Taylor series about

x = 0

for

f (x) = sin 2 x

. (Hint: start with the Taylor series for

sin x

Not correct. Choice (a) is false.

Your answer is correct.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 3

Which one of the following is the Taylor series for

sin x

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

Question 4

Which of the following are true statements?

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.

Your answers are correct

False.
True.
True.
False.

Question 5

\sum_{n = 0}^{\infty} {(- x)}^{n}

is the Taylor series for which function?

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

Not correct. Choice (e) is false.

Question 6

Find the Taylor series about

x = 0

for

f (x) = \frac{1}{{(1 - x)}^{2}}

Your answer is correct.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 7

Several of the following series of numbers converge. Check all those that do.

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 (n + 1)}{2}

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

sin x

evaluated at the point

x = π

. It converges to

sin π = 0

There is at least one mistake.
For example, choice (e) should be false.

The partial sums are

S_{1} = 1, S_{2} = - 1, S_{3} = 2, S_{4} = - 2, S_{5} = 3, S_{6} = - 3

and so on. We see that adding up the first

2 n

terms gives

- n

while adding up the first

2 n - 1

terms gives

n

. Clearly these partial sums never settle down to a limit as

n

approaches infinity, so the series diverges.

Your answers are correct

False. Adding up the first $n$ terms of this series gives $1 + 2 + 3 + \dots \dots + n = \frac{n (n + 1)}{2}$ 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 $sin x$ evaluated at the point $x = π$ . It converges to $sin π = 0$ .
False. The partial sums are $S_{1} = 1, S_{2} = - 1, S_{3} = 2, S_{4} = - 2, S_{5} = 3, S_{6} = - 3$ and so on. We see that adding up the first $2 n$ terms gives $- n$ while adding up the first $2 n - 1$ terms gives $n$ . Clearly these partial sums never settle down to a limit as $n$ approaches infinity, so the series diverges.

Question 8

What is the value of

0.77777777 \dots \dots

Not correct. Choice (a) is false.

Your answer is correct.

Note that

0.77777777 \dots \dots = \frac{7}{10} + \frac{7}{100} + \frac{7}{1000} + \dots \dots = \frac{7}{10} (1 + \frac{1}{10} + \frac{1}{100} + \dots \dots)

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} .

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 9

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.

Your answer is correct

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

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} .

Not correct. You may try again.

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

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

Question 10

Which series equals the Taylor series of

cosh (- x^{2})

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Not correct. Choice (f) is false.

ABN: 15 211 513 464. CRICOS number: 00026A. Phone: +61 2 9351 2222.

Authorised by: Head, School of Mathematics and Statistics.

Contact the University | Disclaimer | Privacy | Accessibility

Sitemap

For questions or comments please contact webmaster@maths.usyd.edu.au

Questions:

right first attempt
right
wrong

a)	$- 2 ∕ 3$	b)	$- 4 ∕ 3$
c)	$4 ∕ 3$	d)	$- 8 ∕ 3$
e)	$2 ∕ 3$

a)	$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n - 1}}{(2 n - 1)!}$	b)	$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n + 1} x^{2 n + 1}}{(2 n + 1)!}$
c)	$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1)!}$	d)	$\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1)!}$

a)		Every function $f (x)$ is equal to its Taylor series for all real $x$ .
b)		There exist functions $f (x)$ which are equal to their Taylor series for all real $x$ .
c)		There exist functions $f (x)$ which are equal to their Taylor series for some, but not all, real numbers $x$ .
d)		A function $f (x)$ can never equal its Taylor series. The Taylor series is only ever an approximation to the function.

a)	$sin x$	b)	$cos x$
c)	$\frac{1}{1 - x}$	d)	$\frac{1}{1 + x}$
e)	$e^{x}$

a)	$\sum_{n = 0}^{\infty} (n + 1) x^{n}$	b)	$\sum_{n = 0}^{\infty} {(- 1)}^{n} (n + 1) x^{n}$
c)	$\sum_{n = 0}^{\infty} n x^{n}$	d)	$\sum_{n = 0}^{\infty} x^{2 n}$
e)	$\sum_{n = 0}^{\infty} - x^{2 n}$

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)	$π - \frac{π^{3}}{3!} + \frac{π^{5}}{5!} - \frac{π^{7}}{7!} + \dots \dots$
e)	$1 - 2 + 3 - 4 + 5 - 6 + \dots \dots$

a)	$\frac{7}{10}$	b)	$\frac{7}{9}$
c)	$\frac{70}{91}$	d)	$\frac{69}{90}$
e)	None of the above

a)	$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{4 n}}{(2 n)!}$	b)	$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n - 1}}{(2 n - 1)!}$
c)	$\sum_{n = 0}^{\infty} \frac{x^{4 n}}{(2 n)!}$	d)	$\sum_{n = 1}^{\infty} \frac{x^{4 n}}{(2 n)!}$
e)	$\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n}}{(2 n)!}$	f)	None of the above.

The University of Sydney - School of Mathematics and Statistics

Quiz 12: Taylor series

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Sitemap