Question 1

Which of the following is the general form of a power series centered at $$x = a$$?

Accepted Answer

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

Question 2

Given the power series $$\sum_{n=0}^{\infty} x^n$$, what is its interval of convergence?

Accepted Answer

$$(-1, 1)$$

Question 3

Suppose a power series $$\sum_{n=0}^{\infty} c_n (x - 2)^n$$ converges at $$x = 5. $$What can you conclude about its convergence?

Accepted Answer

It converges absolutely for all $$x$$ with $$|x - 2| \u003c 3$$

Question 4

Which test is most commonly used to determine the radius of convergence for a power series?

Accepted Answer

Ratio Test

Question 5

If a power series $$\sum_{n=0}^{\infty} c_n (x - a)^n$$ has radius of convergence $$R$$, what happens at $$x = a + R$$?

Accepted Answer

The series may converge or diverge; it must be tested separately.

Question 6

Given $$f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n$$ with radius of convergence $$R$$, what is the interval where $$f(x) is $$differentiable?

Accepted Answer

$$|x - a| \u003c R$$

Question 7

Which of the following statements about the term-by-term differentiation of a power series is correct?

Accepted Answer

The differentiated series has the same radius of convergence as the original series.

Question 8

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

Accepted Answer

$$\infty$$

Question 9

If $$\sum_{n=0}^{\infty} c_n x^n$$ converges for $$x = 2$$ and diverges for $$x = 4$$, what can you say about the radius of convergence $$R$$?

Accepted Answer

$$2 \u003c R \u003c 4$$

Question 10

Which of the following is a valid application of the Series Multiplication Theorem for power series?

Accepted Answer

If $$\sum a_n x^n$$ and $$\sum b_n x^n$$ converge absolutely for $$|x| \u003c R$$, then their product $$\sum c_n x^n$$ converges absolutely for $$|x| \u003c R.$$

Question 11

Suppose $$f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n$$ converges for $$|x - a| \u003c R. $$What is the result of integrating $$f(x)$$ term by term within this interval?

Accepted Answer

The integrated series converges for $$|x - a| \u003c R.$$

Question 12

Which of the following best describes the interval of convergence for the power series $$\sum_{n=0}^{\infty} n! x^n$$?

Accepted Answer

$$x = 0$$ only

Question 13

A power series $$\sum_{n=0}^{\infty} c_n (x - a)^n$$ converges absolutely for $$|x - a|  2. $$What must be checked to fully determine the interval of convergence?

Accepted Answer

Convergence at $$x = a \pm 2$$

Question 14

Which of the following is a real-world application of power series?

Accepted Answer

Approximating solutions to differential equations in physics and engineering.

Question 15

Given the power series $$\sum_{n=0}^{\infty} \frac{(x-1)^n}{2^n}$$, what is its radius of convergence?

Accepted Answer

$$2$$

Question 16

If $$f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n$$ and $$f(x) is $$continuous on $$|x - a| \u003c R$$, what can be said about the sum function $$f(x)$$?

Accepted Answer

It is infinitely differentiable on $$|x - a| \u003c R.$$

Question 17

Which of the following is NOT a possible interval of convergence for a power series?

Accepted Answer

$$[a, b]$$ where $$a \u003c b$$

Question 18

Suppose $$\sum_{n=0}^{\infty} c_n (x - a)^n$$ converges for all $$x. $$What can you say about the coefficients $$c_n$$?

Accepted Answer

$$c_n$$ must decrease rapidly enough for the series to converge for all $$x.$$

Power Series: Definitions, Convergence, and Applications

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