Free Practice Quiz Question List

Power Series and Taylor Expansions Online Quiz Questions

Use this free practice quiz with 20 questions to review Power Series and Taylor Expansions, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Which description correctly identifies a Maclaurin series?

A
A Taylor series centered at x=1
B
A Taylor series centered at x=0
C
A power series with radius 1
D
A geometric series with first term 1

True or false

1 point

True or false: For a power series with radius of convergence R, the endpoints of the interval of convergence must be tested separately.

A
True
B
False

True or false

1 point

True or false: Term-by-term integration of a power series always changes its radius of convergence.

A
True
B
False

Choose one

1 point

What is the interval of convergence of $\displaystyle\sum_{n=0}^{\infty}\frac{(x-2)^n}{(n+1)3^n}$ ?

A
(-1,5)
B
(-1,5]
C
[-1,5)
D
[-1,5]

Choose one

1 point

Which series represents $e^{x^2}$ ?

A
$\displaystyle\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$
B
$\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{(2n)!}$
C
$\displaystyle\sum_{n=0}^{\infty}\frac{x^{n+2}}{n!}$
D
$\displaystyle\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{n!}$

Choose all

1 point

Select all correct statements about term-by-term operations on a power series within its open interval of convergence.

A
Term-by-term differentiation is valid inside the open interval of convergence.
B
Term-by-term operations always preserve convergence or divergence at every endpoint.
C
Term-by-term integration is valid inside the open interval of convergence.
D
Differentiation necessarily doubles the radius of convergence.

Choose all

1 point

Select all statements that correctly describe Taylor's theorem with the Lagrange remainder bound.

A
The remainder depends only on f(a), regardless of n.
B
The Lagrange bound uses a bound on $f^{(n+1)}$ .
C
The bound is valid without any derivative assumptions.
D
The bound has the form $\frac{M}{(n+1)!}|x-a|^{n+1}$ .

Written response

1 point

For $\displaystyle\sum_{n=0}^{\infty}\frac{(x-2)^n}{(n+1)3^n}$ , enter the radius of convergence as a number.

Written response

1 point

What is the standard name for the estimate $\displaystyle |R_n(x)|\le\frac{M}{(n+1)!}|x-a|^{n+1}$ ? Enter the name of the bound.

Fill in the blank

1 point

Complete the statement: In a power series $\displaystyle\sum_{n=0}^{\infty}c_n(x-a)^n$ , the number $a$ is the , and the number $R$ determining $|x-a|<R$ is the .

Fill in the blank

1 point

Complete the statement: For an alternating series whose term magnitudes decrease to zero, bounding the error by the first omitted term is called the .

Open ended

1 point

Find a power series for $\displaystyle\frac{1}{x}$ centered at $x=1$ , and state its interval of convergence. Show how the geometric series is used.

Choose one

1 point

Assume $\displaystyle y(x)=\sum_{n=0}^{\infty}a_nx^n$ in the differential equation $y''+y=0$ . Which recurrence relation results after aligning powers of x?

A
$\displaystyle a_{n+2}=-\frac{a_n}{(n+2)(n+1)}$
B
$\displaystyle a_{n+2}=-\frac{a_{n+1}}{n+2}$
C
$\displaystyle a_{n+2}=\frac{a_n}{(n+2)(n+1)}$
D
$\displaystyle a_{n+2}=-(n+2)(n+1)a_n$

Choose one

1 point

What is the center of the power series $\sum_{n=0}^{\infty}(x-3)^n$ ?

A
0
B
3
C
1
D
-3

Choose one

1 point

What is the interval of convergence of $\sum_{n=0}^{\infty}\frac{(x-2)^n}{(n+1)3^n}$ ?

A
$(-1,5)$
B
$[-1,5]$
C
$[-1,5)$
D
$(-1,5]$

Choose one

1 point

Suppose a power series has radius of convergence $R>0$ . Which statement about term-by-term differentiation and integration is correct?

A
Both operations always double the radius.
B
Differentiation preserves the radius, but integration always changes it.
C
Integration preserves the radius, but differentiation always changes it.
D
Both operations preserve the radius, although endpoint behavior may differ.

Choose one

1 point

Which power-series representation and interval of convergence correctly describe $\frac{1}{2+x}$ about $x=0$ ?

A
$\sum_{n=0}^{\infty}\frac{x^n}{2^n}$ , for $|x|<1$
B
$\sum_{n=0}^{\infty}\frac{(-1)^n x^n}{2^{n+1}}$ , for $|x|<2$
C
$\sum_{n=0}^{\infty}\frac{(-1)^n x^n}{2^n}$ , for $|x|<2$
D
$\sum_{n=0}^{\infty}\frac{x^n}{2^{n+1}}$ , for $|x|<2$

True or false

1 point

True or false: Within the open interval of convergence, term-by-term integration preserves the radius of convergence of a power series, although endpoint behavior may change.

A
True
B
False

Written response

1 point

For the differential equation $y''+y=0$ , a power-series solution satisfies $a_{n+2}=-\frac{a_n}{(n+2)(n+1)}$ . If $a_0=6$ , what is the value of $a_4$ ?

Written response

1 point

The first four terms are used to approximate $\int_0^1 e^{-t^2}\,dt$ with the series $1-\frac13+\frac1{10}-\frac1{42}+\cdots$ . What is the alternating-series upper bound for the error?