Q: Suppose you are given the series $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$. Which of the following is the correct limit to use for the integral test?

$\lim_{t \to \infty} \int_1^t \frac{1}{x^{3/2}} dx$

Question 1

Given the series $\sum_{n=1}^{\infty} a_n$ where $a_n = \ln(2n) - \ln(n+2)$, which of the following is the correct formula for the $n$th partial sum $S_n$?

Accepted Answer

$S_n = \ln(2n) - \ln(n+2) - \ln(2) + \ln(1)$

Question 2

Does the series $\sum_{n=1}^{\infty} (\ln(2n) - \ln(n+2))$ converge or diverge?

Accepted Answer

Diverges, because the partial sums grow without bound

Question 3

Which of the following is the correct formula for the $n$th term $a_n$ of the series $\sum_{n=1}^{\infty} (\ln(2n) - \ln(n+2))$?

Accepted Answer

$a_n = \ln(2n) - \ln(n+2)$

Question 4

Given the series $\sum_{n=1}^{\infty} \left( \frac{(-2)^n}{3^n} \right)$, which of the following statements is true?

Accepted Answer

The series converges because it is a geometric series with $|r| < 1$

Question 5

What is the sum of the series $\sum_{n=1}^{\infty} \left( \frac{(-2)^n}{3^n} \right)$?

Accepted Answer

$-2/3$

Question 6

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

Accepted Answer

Integral Test

Question 7

Evaluate the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$ using the integral test.

Accepted Answer

The series converges because the integral $\int_1^{\infty} \frac{1}{x^{3/2}} dx$ converges

Question 8

For the series $\sum_{n=1}^{\infty} \frac{1}{n^p}$, which value of $p$ guarantees convergence?

Accepted Answer

$p > 1$

Question 9

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

Accepted Answer

The series converges by the integral test

Question 10

Which of the following is the correct application of the integral test to determine the convergence of $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$?

Accepted Answer

$\int_1^{\infty} \frac{1}{x^{3/2}} dx$ converges, so the series converges

Question 11

Suppose you are given the series $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$. Which of the following is the correct limit to use for the integral test?

Accepted Answer

$\lim_{t 	o \infty} \int_1^t \frac{1}{x^{3/2}} dx$

Question 12

Which of the following series is convergent by the integral test?

Accepted Answer

$\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$

Question 13

A medical researcher is analyzing the convergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ to model the decay of a drug in the bloodstream. For which value of $p$ does the model predict the drug concentration will eventually reach zero?

Accepted Answer

$p > 1$

Question 14

Which of the following is a correct justification for using the integral test on $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$?

Accepted Answer

The function $f(x) = \frac{1}{x^{3/2}}$ is continuous, positive, and decreasing for $x \geq 1$

Question 15

A chemical engineer is studying a reaction modeled by the series $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$. What does the convergence of this series imply about the total amount of reactant consumed over time?

Accepted Answer

The total amount consumed is finite

Question 16

Which of the following is the correct evaluation of $\lim_{n 	o \infty} \arctan n - \arctan e$?

Accepted Answer

$\frac{\pi}{2} - \arctan e$

Question 17

Given the series $\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}$, which of the following statements is true?

Accepted Answer

The series converges by the integral test

Series Convergence and Partial Sums in Calculus II

Study Guide - Practice Questions

Study Guide - Flashcards