Q: Given $a_n = \frac{1}{n^2}$, what is an upper bound for the error when approximating $\sum_{n=1}^\infty a_n$ by the sum of the first $k$ terms?

$\int_k^\infty \frac{1}{x^2}\,dx$

Q: Which of the following statements is true about the Limit Comparison Test?

It compares the limit of the ratio of terms of two series.

Q: Consider the series $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$. Using the Direct Comparison Test, which statement is correct?

$\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges.

Q: If $a_n = \frac{1}{n^p}$ and $p = 1$, what can be said about the convergence of $\sum_{n=1}^\infty a_n$?

The series diverges for $p = 1$.

Q: Which of the following is a necessary condition for applying the Integral Test to a series $\sum_{n=1}^\infty a_n$?

The function $f(x)$ corresponding to $a_n$ must be continuous, positive, and decreasing for $x \geq 1$.

Question 1

Suppose $f(x)$ is a continuous, positive, decreasing function for $x \geq 1$ and $\int_1^\infty f(x)\,dx$ is convergent. What can be said about the convergence or divergence of $\sum_{n=1}^\infty f(n)$?

Accepted Answer

The series $\sum_{n=1}^\infty f(n)$ converges because the integral $\int_1^\infty f(x)\,dx$ converges.

Question 2

Given $f(x)$ is continuous, positive, and decreasing for $x \geq 1$, which of the following best describes the relationship between $\sum_{n=1}^\infty f(n)$ and $\int_1^\infty f(x)\,dx$?

Accepted Answer

$\sum_{n=1}^\infty f(n)$ and $\int_1^\infty f(x)\,dx$ converge or diverge together.

Question 3

If $\int_1^\infty f(x)\,dx$ is finite, what can be said about $\sum_{n=1}^\infty f(n)$?

Accepted Answer

$\sum_{n=1}^\infty f(n)$ converges.

Question 4

Which test is most appropriate to determine the convergence of $\sum_{n=1}^\infty \frac{1}{n^p}$ for $p > 1$?

Accepted Answer

The Integral Test.

Question 5

For the series $\sum_{n=1}^\infty \frac{1}{n^2}$, which of the following is true?

Accepted Answer

The series converges because $\int_1^\infty \frac{1}{x^2}\,dx$ is finite.

Question 6

Suppose you use the Integral Test to estimate the sum $S$ of a convergent series $\sum_{n=1}^\infty a_n$ and want the error to be less than $0.01$. Which inequality should you solve for $k$?

Accepted Answer

$\int_k^\infty f(x)\,dx < 0.01$

Question 7

Given $a_n = \frac{1}{n^2}$, what is an upper bound for the error when approximating $\sum_{n=1}^\infty a_n$ by the sum of the first $k$ terms?

Accepted Answer

$\int_k^\infty \frac{1}{x^2}\,dx$

Question 8

Which of the following statements is true about the Limit Comparison Test?

Accepted Answer

It compares the limit of the ratio of terms of two series.

Question 9

Consider the series $\sum_{n=1}^\infty \frac{1}{n}$ and $\sum_{n=1}^\infty \frac{1}{n^2}$. Using the Direct Comparison Test, which statement is correct?

Accepted Answer

$\sum_{n=1}^\infty \frac{1}{n}$ diverges and $\sum_{n=1}^\infty \frac{1}{n^2}$ converges.

Question 10

If $a_n = \frac{1}{n^p}$ and $p = 1$, what can be said about the convergence of $\sum_{n=1}^\infty a_n$?

Accepted Answer

The series diverges for $p = 1$.

Question 11

Which of the following is a necessary condition for applying the Integral Test to a series $\sum_{n=1}^\infty a_n$?

Accepted Answer

The function $f(x)$ corresponding to $a_n$ must be continuous, positive, and decreasing for $x \geq 1$.

Question 12

Suppose $f(x)$ is continuous, positive, and decreasing for $x \geq 1$. Which of the following inequalities is always true for $n \geq 1$?

Accepted Answer

$\int_n^{n+1} f(x)\,dx \leq f(n)$

Question 13

If $\sum_{n=1}^\infty a_n$ converges by the Integral Test, which of the following gives a lower bound for the error in approximating the sum by the first $k$ terms?

Accepted Answer

$\int_{k+1}^\infty f(x)\,dx$

Question 14

Which of the following is a correct application of the Limit Comparison Test for $\sum_{n=1}^\infty \frac{n}{n^3+1}$?

Accepted Answer

Compare with $\sum_{n=1}^\infty \frac{1}{n^2}$.

Question 15

If $\lim_{n 	o \infty} \frac{a_n}{b_n} = L$ where $0 < L < \infty$, what does the Limit Comparison Test tell us?

Accepted Answer

Both $\sum a_n$ and $\sum b_n$ either converge or diverge together.

Question 16

A medical researcher models the decay of a drug in the bloodstream by $f(x) = \frac{1}{x^2}$ for $x \geq 1$. If the total exposure is measured by $\int_1^\infty f(x)\,dx$, what is the value of this integral?

Accepted Answer

$1$

Question 17

Which of the following series converges?

Accepted Answer

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

Question 18

A chemist wants to estimate the total amount of a substance released over time using $\sum_{n=1}^\infty \frac{1}{n^3}$. Which test should be used to determine convergence?

Accepted Answer

Integral Test

Convergence of Series and Integral Tests in Calculus

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