Q: Evaluate the convergence of the series $\sum\limits_{n=1}^{\infty} \frac{n^2}{5^n}$, which could represent the probability of a rare event in a genetics experiment.

It converges by the Ratio Test.

Q: Which of the following series is absolutely convergent but not conditionally convergent?

$\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n^2}$

Q: Which of the following is the Taylor series for $f(x) = e^x$ about $x = 0$?

$\sum\limits_{n=0}^{\infty} \frac{x^n}{n!}$

Q: Which of the following series diverges?

$\sum\limits_{n=1}^{\infty} \frac{1}{n}$

Q: Given the sequence $a_n = \frac{1}{4n}$, which of the following is true?

It is decreasing, bounded below by 0, and converges to 0.

Q: Which of the following is the correct power series representation for $g(x) = \frac{1}{1 + x^2}$?

$\sum\limits_{n=0}^{\infty} (-1)^n x^{2n}$

Q: A population model uses the series $\sum\limits_{n=1}^{\infty} \frac{1}{n \ln(n)}$. What can be said about its convergence?

It diverges by the Integral Test.

Q: Which of the following is the correct Taylor series for $\sin(x)$ about $x = 0$?

$\sum\limits_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Question 1

Which of the following tests is most appropriate to determine the convergence of the series $\sum\limits_{n=1}^{\infty} \frac{2^n + 4^n}{7^n}$?

Accepted Answer

The Ratio Test

Question 2

Consider the series $\sum\limits_{n=1}^{\infty} \frac{n^3}{n!}$. Which statement is true?

Accepted Answer

The series converges by the Ratio Test.

Question 3

Which of the following best describes the convergence of the series $\sum\limits_{n=1}^{\infty} \frac{1}{2^n}$?

Accepted Answer

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

Question 4

Suppose a medical researcher is analyzing the convergence of $\sum\limits_{n=2}^{\infty} \frac{1}{n(\ln(n))^3}$ to model the decay of a drug in the bloodstream. Which test should be used?

Accepted Answer

The Integral Test

Question 5

Evaluate the convergence of the series $\sum\limits_{n=1}^{\infty} \frac{n^2}{5^n}$, which could represent the probability of a rare event in a genetics experiment.

Accepted Answer

It converges by the Ratio Test.

Question 6

Given the sequence $a_n = \frac{3n^2 - 1}{10n + 5n^2}$, what is $\lim\limits_{n 	o \infty} a_n$?

Accepted Answer

$\frac{3}{5}$

Question 7

Which of the following series is absolutely convergent but not conditionally convergent?

Accepted Answer

$\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{n^2}$

Question 8

A pharmaceutical company models the concentration of a drug using the power series $\sum\limits_{n=1}^{\infty} \frac{(x-6)^n}{n^n}$. What is the radius of convergence?

Accepted Answer

$\infty$

Question 9

Which of the following is the Taylor series for $f(x) = e^x$ about $x = 0$?

Accepted Answer

$\sum\limits_{n=0}^{\infty} \frac{x^n}{n!}$

Question 10

Which of the following series diverges?

Accepted Answer

$\sum\limits_{n=1}^{\infty} \frac{1}{n}$

Question 11

Given the sequence $a_n = \frac{1}{4n}$, which of the following is true?

Accepted Answer

It is decreasing, bounded below by 0, and converges to 0.

Question 12

Which of the following is the correct power series representation for $g(x) = \frac{1}{1 + x^2}$?

Accepted Answer

$\sum\limits_{n=0}^{\infty} (-1)^n x^{2n}$

Question 13

A population model uses the series $\sum\limits_{n=1}^{\infty} \frac{1}{n \ln(n)}$. What can be said about its convergence?

Accepted Answer

It diverges by the Integral Test.

Question 14

Which of the following is the correct Taylor series for $\sin(x)$ about $x = 0$?

Accepted Answer

$\sum\limits_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Question 15

Which of the following series is conditionally convergent but not absolutely convergent?

Accepted Answer

$\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$

Question 16

A physicist is analyzing the power series $\sum\limits_{n=1}^{\infty} \frac{2^n (4x - 6)^n}{n^n}$. What is the radius of convergence?

Accepted Answer

$\infty$

Question 17

Which of the following is the correct strategy to determine the convergence of $\sum\limits_{n=1}^{\infty} \frac{n^2 + 2n}{n^3 + 3n^2}$?

Accepted Answer

Use the Limit Comparison Test with $\frac{1}{n}$.

Question 18

Which of the following series converges to a finite sum?

Accepted Answer

$\sum\limits_{n=1}^{\infty} \frac{1}{n^2}$

Question 19

Given the sequence $a_n = \frac{2n^2 - 1}{n}$, what is $\lim\limits_{n 	o \infty} a_n$?

Accepted Answer

$\infty$

Question 20

Which of the following is the correct Taylor series for $\ln(x)$ about $x = 2$?

Accepted Answer

$\sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} (x-2)^n 2^{-n}$

Convergence and Divergence of Series and Sequences; Power Series and Taylor Series

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