AP Calculus BC Practice Quiz: Radius and Interval of Convergence of Power Series

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 14 questions to check your progress.

Question 1 of 14

A series is given by the form $\sum_{n=0}^{\infty} a_n (x-r)^n$, where $n$ is a non-negative integer, $\{a_n\}$ is a sequence of real numbers, and $r$ is a real number. What is this type of series called?

A)A geometric seriesB)A p-seriesC)A power seriesD)An alternating series

All Questions (14)

A) A geometric series

B) A p-series

C) A power series

D) An alternating series

Correct Answer: C

Based on the provided content, a series of the form $\sum_{n=0}^{\infty} a_n (x-r)^n$ is defined as a power series. [cite: 3232]

In the power series $\sum_{n=0}^{\infty} 5 (x-2)^n$, what is the value of $r$, the center of the series?

A) 0

B) 2

C) 5

D) n

Correct Answer: B

The general form of a power series is $\sum_{n=0}^{\infty} a_n (x-r)^n$. Comparing this to the given series $\sum_{n=0}^{\infty} 5 (x-2)^n$, the value of $r$ is 2. [cite: 3232]

Which of the following is a primary application of the ratio test in the context of power series?

A) To find the sum of the series.

B) To determine the radius of convergence.

C) To test for convergence at the endpoints of the interval.

D) To identify the function represented by the series.

Correct Answer: B

The provided content explicitly states that 'The ratio test can be used to determine the radius of convergence of a power series.' [cite: 3234]

A power series is found to have a radius of convergence $R=3$ and is centered at $r=1$. Which of the following steps is necessary to determine the full interval of convergence?

A) Applying the ratio test again to the endpoints.

B) Testing the series for convergence at $x=-2$ and $x=4$.

C) Integrating the series term-by-term.

D) The interval is always open, so no further steps are needed.

Correct Answer: B

The radius of convergence $R=3$ centered at $r=1$ identifies an open interval of convergence $(-2, 4)$. According to the content, 'it is necessary to test both endpoints of the interval to determine the interval of convergence.' The endpoints are $1-3=-2$ and $1+3=4$. [cite: 3235]

According to the properties of power series, which of the following describes the possible sets on which a power series can converge?

A) It always converges for all real numbers.

B) It converges only at two distinct points.

C) It converges at a single point or on an interval.

D) It converges on a set of disconnected points.

Correct Answer: C

The content states that 'If a power series converges, it either converges at a single point or has an interval of convergence.' [cite: 3233]

A power series $\sum_{n=0}^{\infty} a_n x^n$ has a radius of convergence of $R=5$. What is the radius of convergence of the series obtained by term-by-term differentiation, $\sum_{n=1}^{\infty} n a_n x^{n-1}$?

A) $R=0$

B) $R=1/5$

C) $R=5$

D) $R=\infty$

Correct Answer: C

The content specifies that 'The radius of convergence of a power series obtained by term-by-term differentiation...is the same as the radius of convergence of the original power series.' Therefore, the new radius is also $R=5$. [cite: 3237]

The power series representation for a function $f(x)$ has a radius of convergence $R$. A new series is formed by taking the term-by-term integral of the original series. What is the radius of convergence of this new series?

A) $R$

B) $R^2$

C) $R-1$

D) The radius cannot be determined without knowing the series.

Correct Answer: A

The provided content states that 'The radius of convergence of a power series obtained by...term-by-term integration is the same as the radius of convergence of the original power series.' [cite: 3237]

After using the ratio test to find that a power series centered at $x=0$ has a radius of convergence of $R=2$, why must the values $x=-2$ and $x=2$ be tested separately?

A) The ratio test is only valid for the interior of the interval.

B) The ratio test always yields a limit of 1 at the endpoints, which is inconclusive.

C) The series might converge or diverge at the endpoints, and the ratio test does not provide this information.

D) To ensure the center of convergence was calculated correctly.

Correct Answer: C

The ratio test determines an open interval of convergence. The behavior at the endpoints is not determined by the ratio test (it typically yields an inconclusive result of 1). Therefore, as stated in the content, 'it is necessary to test both endpoints of the interval to determine the interval of convergence.' This is because the series could converge or diverge at these specific points. [cite: 3235]

A power series $\sum_{n=0}^{\infty} a_n (x-4)^n$ is known to converge at $x=7$ and diverge at $x=9$. Which of the following statements must be true?

A) The radius of convergence is exactly $R=3$.

B) The series must converge at $x=1$.

C) The radius of convergence $R$ satisfies $3 \le R \le 5$.

D) The series must diverge at $x=-1$.

Correct Answer: C

The series is centered at $r=4$. Since it converges at $x=7$, the distance from the center is $|7-4|=3$. This means the radius of convergence $R$ must be at least 3, so $R \ge 3$. Since it diverges at $x=9$, the distance from the center is $|9-4|=5$. This means the radius of convergence $R$ cannot be greater than 5, so $R \le 5$. Combining these, we get $3 \le R \le 5$. [cite: 3231, 3235]

If a power series $\sum_{n=0}^{\infty} a_n x^n$ converges to a function $f(x)$ on the interval $(-2, 2)$, what is the relationship between the series and the function?

A) The series is an approximation of $f(x)$ but not necessarily its Taylor series.

B) The series is the Taylor series for $f(x)$ centered at $x=0$.

C) The series is the Maclaurin series for $f(x)$ only if all $a_n$ are positive.

D) The relationship cannot be determined without knowing the coefficients $a_n$.

Correct Answer: B

The content states that 'If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval.' Since the series converges on $(-2, 2)$, the radius of convergence is $R=2$, which is positive. A Taylor series centered at $x=0$ is also known as a Maclaurin series. [cite: 3236]

The interval of convergence for a power series $S$ is found to be $[-3, 1)$. Let $S'$ be the series obtained by term-by-term differentiation of $S$. What can be concluded about the interval of convergence for $S'$?

A) It is also $[-3, 1)$.

B) It is $(-3, 1)$.

C) The radius of convergence is 2, but the interval could be different.

D) The radius of convergence is 4.

Correct Answer: C

The original interval $[-3, 1)$ has a center at $r = (-3+1)/2 = -1$ and a radius of $R = (1 - (-3))/2 = 2$. According to the content, the radius of convergence for the differentiated series $S'$ will be the same, so $R=2$. However, term-by-term differentiation can change the convergence behavior at the endpoints. The original series converged at $x=-3$ but not at $x=1$. The new series $S'$ must be re-tested at its endpoints, $x=-3$ and $x=1$. Therefore, we only know for sure that the radius is 2, but the interval itself must be re-evaluated. [cite: 3237, 3235]

For the power series $\sum_{n=0}^{\infty} \frac{(x+1)^n}{n!}$, which expression represents the sequence of coefficients $\{a_n\}$?

A) $a_n = n!$

B) $a_n = (x+1)^n$

C) $a_n = \frac{1}{n!}$

D) $a_n = x+1$

Correct Answer: C

The general form of a power series is $\sum_{n=0}^{\infty} a_n (x-r)^n$. In the given series, the term $(x-r)^n$ corresponds to $(x+1)^n = (x-(-1))^n$. The remaining part of the term is the coefficient $a_n$. Therefore, $a_n = \frac{1}{n!}$. [cite: 3232]

Which sequence of steps correctly outlines the process to determine the interval of convergence for a power series?

A) Test the endpoints, then use the ratio test to find the radius.

B) Use the ratio test to find the open interval of convergence, then test the endpoints.

C) Integrate the series, find the interval for the new series, and use that for the original.

D) Find the sum of the series, then determine the domain of the resulting function.

Correct Answer: B

The standard procedure is to first use a convergence test like the ratio test to find the radius of convergence, which defines an open interval. After that, it is necessary to test the two endpoints of this interval individually to determine the complete interval of convergence. [cite: 3231, 3234, 3235]

A power series $\sum_{n=0}^{\infty} a_n (x-5)^n$ is found to converge only at $x=5$. What is the radius of convergence, $R$?

A) $R=0$

B) $R=1$

C) $R=5$

D) $R=\infty$

Correct Answer: A

The content states that a power series can converge at a single point. A power series always converges at its center, which is $x=5$ in this case. If it converges *only* at this single point, the interval of convergence has zero width, meaning the radius of convergence is $R=0$. [cite: 3233]