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 $$p \u003e 1$$?

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

$$\int_k^\infty f(x)\,dx \u003c 0.01$$

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 $$0 \u003c L \u003c \infty$$, 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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