Power Series and Taylor Series: Online Learning Quizzes

Question 1

The radius of convergence of the power series centered at z0 with coefficients {a_n} is given by:

Accepted Answer

lim sup |a_n|^(1/n)

Answer

lim inf |a_n|^(1/n)

Answer

lim sup |z_n - z0|^(1/n)

Answer

lim inf |z_n - z0|^(1/n)

Question 2

Which of the following tests is used specifically to determine the convergence of a power series centered at (z_0)?

Accepted Answer

Ratio Test

Answer

Root Test

Answer

Comparison Test

Answer

Integral Test

Question 3

For the function (f(z) = rac{1}{(z-1)^3}), determine the order of its pole at (z=1).

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 4

Which condition is sufficient to guarantee that a power series represents an entire function?

Accepted Answer

The radius of convergence of the power series is (infty).

Answer

The power series has a finite number of non-zero coefficients.

Answer

The coefficients of the power series alternate in sign.

Answer

The coefficients of the power series are all positive.

Question 5

Which of the following is NOT a field where complex analysis finds applications in engineering?

Accepted Answer

Statistics

Answer

Fluid dynamics

Answer

Heat transfer

Answer

Electromagnetism

Question 6

State the uniqueness property of power series representations and illustrate it with an example.

Accepted Answer

Within its radius of convergence, a power series representation is unique.

Answer

A power series representation is unique for all values of the independent variable.

Answer

A power series representation is unique if it has a finite number of non-zero terms.

Answer

A power series representation is unique if its coefficients are all rational numbers.

Question 7

Which of the following is a sufficient condition for a power series ∑(n=0 to ∞) a_n(x - c)^n to converge?

Accepted Answer

The limit of the absolute value of the ratio of successive terms approaches 0 as n approaches infinity

Answer

The limit of the absolute value of the ratio of successive terms approaches infinity as n approaches infinity

Answer

The limit of the absolute value of the ratio of successive terms approaches 1 as n approaches infinity

Answer

The limit of the absolute value of the ratio of successive terms is greater than 1 as n approaches infinity

Question 8

Under what condition is the Taylor series of a function f(x) about x = a unique?

Accepted Answer

f(x) has a continuous derivative at x = a

Answer

f(x) has a continuous second derivative at x = a

Answer

f(x) has a continuous third derivative at x = a

Answer

f(x) is differentiable at x = a

Question 9

The radius of convergence of the power series ∑(n=0 to ∞) (x - 2)^n is:

Accepted Answer

2

Answer

0

Answer

1

Answer

∞

Question 10

Which of the following is the Laurent series representation?

Accepted Answer

∑(n=-∞ to ∞) a_n(z - z_0)^n

Answer

∑(n=0 to ∞) a_n(z - z_0)^n

Answer

∑(n=1 to ∞) a_n(z - z_0)^n

Answer

∑(n=-∞ to -1) a_n(z - z_0)^n

Question 11

In the Laurent series expansion of f(z) = 1/(z - 1)(z - 2) about z = 2, what is the coefficient of (z - 2)^3?

Accepted Answer

-1/8

Answer

1/8

Answer

-1/4

Answer

1/4

Question 12

Which of the following is not a property of power series?

Accepted Answer

May not converge for all values of the variable

Answer

Can be differentiated or integrated term by term

Answer

Has a radius of convergence

Answer

Can represent a wide variety of functions

Question 13

The Taylor series expansion of f(x) = sin(x) about x = 0 is:

Accepted Answer

∑(n=0 to ∞) (-1)^n x^(2n+1)/(2n+1)!

Answer

∑(n=0 to ∞) x^n/n!

Answer

∑(n=0 to ∞) (-1)^n x^n/(2n)!

Answer

∑(n=0 to ∞) x^(2n)/(2n)!

Question 14

Compute the Taylor series expansion for f(x) = sin(x) about the point x = π/2.

Accepted Answer

∑(n=0 to ∞) (-1)^n (x - π/2)^(2n + 1) / (2n + 1)!

Answer

∑(n=0 to ∞) (-1)^n (x - π/2)^n / n!

Answer

∑(n=0 to ∞) (x - π/2)^n / n!

Answer

∑(n=0 to ∞) (x - π/2)^(2n) / (2n)!

Question 15

Which of the following functions is not analytic?

Accepted Answer

|z|

Answer

e^z

Answer

z^2

Answer

sin(z)

Question 16

Which of the following is a precise formula for the radius of convergence of a power series centered at z0?

Accepted Answer

The distance from z0 to its nearest singularity

Answer

The sum of the absolute values of the coefficients of the series

Answer

The largest value of the denominator of the coefficients of the series

Answer

The smallest value of the numerator of the coefficients of the series

Question 17

In the context of power series, what is the significance of the radius of convergence?

Accepted Answer

It determines the interval where the series converges absolutely.

Answer

It is the value at which the series diverges.

Answer

It is always equal to the interval of convergence.

Answer

It is only defined for power series with positive coefficients.

Question 18

What is the Maclaurin series for sin(x)?

Accepted Answer

x - x^3/3! + x^5/5! - x^7/7! + ...

Answer

2x - x^3/3 + x^5/5 - x^7/7 + ...

Answer

1 - x^2/2! + x^4/4! - x^6/6! + ...

Answer

x^2/2 - x^4/4 + x^6/6 - x^8/8 + ...

Question 19

What is the residue of the function 1/(z^2+1) at z = i?

Accepted Answer

1/2i

Answer

2i

Answer

1/2

Answer

-1/2i

Question 20

Which test can be used to determine if a power series represents an analytic function?

Accepted Answer

Cauchy-Riemann equations

Answer

Convergence tests

Answer

Ratio test

Answer

Uniform convergence test

Question 21

Which function has a singular point at z = 0?

Accepted Answer

1/z

Answer

z^2

Answer

e^z

Answer

sin(z)

Question 22

What is the order of the pole of the function 1/(z-2)^3 at z = 2?

Accepted Answer

3

Answer

∞

Answer

2

Answer

1

Question 23

Which of the following accurately describes the Cauchy-Hadamard theorem?

Accepted Answer

It determines the radius of convergence of a power series using the superior limit of the absolute values of its coefficients.

Answer

It asserts that the sum of a convergent power series is an analytic function.

Question 24

What is an essential singularity of a function?

Accepted Answer

A point where the function cannot be locally represented by a power series.

Answer

A point where the function has a pole of order greater than 1.

Answer

A point where the function has a branch cut.

Question 25

Which of the following is the Taylor series for the function f(x) = e^x?

Accepted Answer

∑[n=0 to ∞] (x^n)/n!

Answer

∑[n=0 to ∞] (x - 1)^n/n!

Answer

∑[n=0 to ∞] x^n

Answer

∑[n=1 to ∞] (x^n)/n^2

Question 26

Which of the following is a necessary condition for a power series to have a non-zero radius of convergence?

Accepted Answer

The limit of the absolute value of the coefficients as n approaches infinity must be zero.

Answer

The coefficients must all be positive.

Answer

The series must be alternating.

Question 27

Which of the following functions possesses a non-isolated singularity at z = 0?

Accepted Answer

f(z) = 1/z^2

Answer

f(z) = sin(z)

Answer

f(z) = z^3

Answer

f(z) = e^z

Question 28

Which of the following is a sufficient condition for a power series to converge uniformly on a compact set?

Accepted Answer

The series converges absolutely for all values of the variable within the compact set.

Answer

The coefficients of the series are all positive.

Answer

The series converges uniformly on a larger compact set.

Question 29

Calculate the residue of the function f(z) = e^z/(z - 1)^2 at the point z = 1.

Accepted Answer

e

Answer

1

Answer

0

Answer

-e

Question 30

What is the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!} $?

Accepted Answer

$\infty$

Answer

2

Answer

0

Answer

1

Question 31

Which of the following statements about the Taylor series of a function f(x) centered at x = a is **FALSE**?

Accepted Answer

The Taylor series of f(x) converges to f(x) for all values of x.

Answer

The Taylor series of f(x) may not converge to f(x) for all values of x.

Answer

The Taylor series of f(x) can be used to approximate f(x) near x = a.

Answer

The Taylor series of f(x) is unique.

Question 32

Find the Taylor series expansion of f(x) = e^x centered at x = 0.

Accepted Answer

$\sum_{n=0}^{\infty} \frac{x^n}{n!}$

Answer

$\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}$

Answer

$\sum_{n=0}^{\infty} x^n$

Answer

$\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n!}$

Question 33

What is the Laurent series expansion of f(z) = \frac{1}{z(z-1)} about z = 1?

Accepted Answer

$\sum_{n=0}^{\infty} \frac{(z-1)^n}{1^n} + \sum_{n=1}^{\infty} \frac{1}{(z-1)^n}$

Answer

$\sum_{n=0}^{\infty} \frac{(z-1)^n}{n!}$

Answer

$\sum_{n=0}^{\infty} (z-1)^n$

Answer

$\sum_{n=0}^{\infty} \frac{(-1)^n (z-1)^n}{n!}$

Question 34

Which of the following is a necessary condition for a power series to converge at a point x?

Accepted Answer

The limit of the terms of the power series as n approaches infinity must be zero.

Answer

The power series must have a finite number of terms.

Answer

The radius of convergence must be infinite.

Answer

The power series must be a Taylor series.

Question 35

Which of the following functions can be represented by a power series centered at x = 0 with a radius of convergence of 1?

Accepted Answer

f(x) = \frac{1}{1+x}$

Answer

f(x) = e^x

Answer

f(x) = cos(x)

Answer

f(x) = sin(x)

Question 36

Which of the following is the general form of a power series centered at (c)?

Accepted Answer

∑_(n=0)^∞ a[n] * (z - c)^n

Answer

∑_(n=1)^∞ a[n] * (z - c)^n

Answer

∑_(n=1)^∞ a[n] * (z + c)^n

Answer

∑_(n=0)^∞ a[n] * (z + c)^n

Question 37

Find the interval of convergence for the power series ∑_(n=0)^∞ (z - 2)^n / n?

Accepted Answer

(1, 3)

Answer

[-2, 2]

Answer

(-1, 1)

Answer

(-∞, ∞)

Question 38

Determine the Taylor series expansion of e^z centered at z = 0.

Accepted Answer

∑_(n=0)^∞ (z^n) / n!

Answer

∑_(n=1)^∞ (z^n) / n!

Answer

∑_(n=0)^∞ z^n

Answer

∑_(n=1)^∞ z^n / n

Question 39

Find the Laurent series expansion of the function f(z) = 1 / (z - 1) centered at z = 1?

Accepted Answer

∑_(n=-1)^∞ z^n

Answer

∑_(n=1)^∞ z^n

Answer

∑_(n=-1)^∞ 1 / z^n

Answer

∑_(n=0)^∞ z^n

Question 40

Calculate the residue of the function f(z) = e^z / (z - 1) at z = 1.

Accepted Answer

e

Answer

0

Answer

-e

Answer

1

Question 41

What is the radius of convergence of the power series ∑_(n=0)^∞ (x^n)/n! ?

Accepted Answer

∞

Answer

2

Answer

1

Answer

0

Question 42

Identify the function that can be represented by a power series centered at z = 0 with a radius of convergence of 1.

Accepted Answer

1/(1-z)

Answer

e^z

Answer

sin(z)

Question 43

Which of the following statements accurately describes the uniqueness of Taylor series representations?

Accepted Answer

If two functions have the same Taylor series representation, they are equal within the radius of convergence.

Answer

Every function can be represented by a unique Taylor series.

Question 44

Select the test that can be used to determine the radius of convergence of a power series.

Accepted Answer

Ratio Test

Answer

Integral Test

Answer

Comparison Test

Answer

Limit Comparison Test

Question 45

Which statement correctly describes the convergence of a Laurent series?

Accepted Answer

A Laurent series converges within an annulus.

Answer

A Laurent series converges within a disk.

Answer

A Laurent series converges on the entire complex plane.

Answer

A Laurent series may not converge.

Question 46

Find the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{z^n}{n!}$.

Accepted Answer

$\infty$

Answer

0

Answer

2

Answer

1

Question 47

Which of the following statements about the power series $\sum_{n=0}^{\infty} a_n (z-z_0)^n$ is TRUE?

Accepted Answer

The power series converges absolutely for all values of (z) within its radius of convergence.

Answer

The power series converges uniformly on any closed disk centered at (z_0) with radius less than the radius of convergence.

Answer

The power series converges conditionally for some values of (z) within its radius of convergence.

Question 48

Find the Taylor series expansion of the function $f(z) = e^z$ about $z_0 = 0$.

Accepted Answer

$\sum_{n=0}^{\infty} \frac{z^n}{n!}$

Answer

$\sum_{n=0}^{\infty} z^n$

Answer

$\sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}$

Answer

$\sum_{n=0}^{\infty} \frac{(-1)^n z^n}{n!}$

Question 49

Find the Taylor series expansion of the function $f(z) = \frac{1}{1-z}$ about $z_0 = 0$.

Accepted Answer

$\sum_{n=0}^{\infty} z^n$

Answer

$\sum_{n=0}^{\infty} \frac{z^{2n}}{(2n)!}$

Answer

$\sum_{n=0}^{\infty} \frac{z^n}{n!}$

Answer

$\sum_{n=0}^{\infty} (-1)^n z^n$

Question 50

Find the Laurent series expansion of the function $f(z) = \frac{1}{z(z-1)}$ about $z_0 = 0$.

Accepted Answer

$\sum_{n=0}^{\infty} z^n - \sum_{n=1}^{\infty} \frac{1}{z^n}$

Answer

$\sum_{n=0}^{\infty} z^n + \sum_{n=1}^{\infty} \frac{1}{z^n}$

Answer

$\sum_{n=0}^{\infty} z^{2n} - \sum_{n=1}^{\infty} \frac{1}{z^{2n}}$

Question 51

Which statement correctly describes the uniqueness of Taylor series?

Accepted Answer

A function has a unique Taylor series expansion about a point if the function is analytic in a neighborhood of that point.

Answer

A function always has a unique Taylor series expansion about any point.

Question 52

Find the interval of convergence for the power series $\sum_{n=0}^{\infty} \frac{z^n}{2^n}$.

Accepted Answer

(-2, 2)

Answer

(-\infty, \infty)

Answer

[-2, 2]

Answer

(-1, 1)

Question 53

Find the residue of the function $f(z) = \frac{1}{z^2(z-1)}$ at the singularity $z = 1$.

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 54

Which series converges to the function $f(z) = \frac{1}{(1+z)^2}$ for $|z| < 1$?

Accepted Answer

$\sum_{n=0}^{\infty} (-1)^n (n+1) z^n$

Answer

$\sum_{n=0}^{\infty} (-1)^n z^n$

Answer

$\sum_{n=0}^{\infty} z^n$

Answer

$\sum_{n=0}^{\infty} (n+1) z^n$

Question 55

Evaluate the integral $\oint_{|z|=2} \frac{e^z}{z^2} dz$ using Cauchy's Residue Theorem.

Accepted Answer

2\pi i

Answer

0

Answer

-2\pi i

Answer

4\pi i