Q: Which of the following is the correct process for the Root Test?

Compute $$\lim\limits_{n \to \infty} \sqrt[n]{|a_n|}$$ and analyze the result.

Question 1

Which of the following series converges by the Ratio Test?

Accepted Answer

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

Question 2

Consider the series $$\sum\limits_{n=2}^{\infty} \frac{1}{n \ln(n)}. $$Which test is most appropriate to determine its convergence?

Accepted Answer

Integral Test

Question 3

Find the sum of the series $$\sum\limits_{n=1}^{\infty} \left( \frac{1}{2^n} - \frac{1}{3^n} \right).$$

Accepted Answer

Question 4

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

Accepted Answer

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

Question 5

For which values of $$p$$ does the series $$\sum\limits_{n=1}^{\infty} \frac{1}{n^p}$$ converge?

Accepted Answer

$$p \u003e 1$$

Question 6

Which of the following is the correct process for the Root Test?

Accepted Answer

Compute $$\lim\limits_{n 	o \infty} \sqrt[n]{|a_n|}$$ and analyze the result.

Question 7

Given the power series representation $$g(x) = \frac{4}{2 - x^2$}$$ centered at $$0$$, what is its interval of convergence?

Accepted Answer

$$|x| \u003c \sqrt{2}$$

Question 8

Which of the following series is conditionally convergent?

Accepted Answer

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

Question 9

A medical researcher models the concentration of a drug in the bloodstream as $$C(t) = \sum\limits_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$}{n} t^n$$ for $$|t| \u003c 1. $$What is the function represented by this series?

Accepted Answer

$$\ln(1 + t)$$

Question 10

Which of the following is NOT a necessary condition for the Direct Comparison Test?

Accepted Answer

The series must be alternating.

Question 11

A chemical reaction rate is modeled by the series $$\sum\limits_{n=1}^{\infty} \frac{n$$^{2n}$$}{4n! + 1}. $$Which test is most appropriate to determine convergence?

Accepted Answer

Ratio Test

Question 12

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

Accepted Answer

It compares the limit of the ratio of terms of two series as $$n 	o \infty.$$

Question 13

Which of the following series diverges by the $$nth $$Term Test?

Accepted Answer

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

Question 14

Which of the following is the correct process for the Alternating Series Test?

Accepted Answer

Check if the terms decrease and the limit of the terms is zero.

Question 15

A population model uses the series $$\sum\limits_{n=1}^{\infty} \frac{2^n}{n^2$}. $$Which test is most appropriate to determine convergence?

Accepted Answer

Ratio Test

Question 16

Which of the following is the correct process for the Geometric Series Test?

Accepted Answer

Check if $$|r| \u003c 1$$ for $$\sum ar^n.$$

Question 17

Which of the following series converges absolutely?

Accepted Answer

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

Question 18

Which of the following is the correct process for the $$p-$$Series Test?

Accepted Answer

Check if $$p \u003e 1$$ for $$\sum\limits_{n=1}^{\infty} \frac{1}{n^p}.$$

Question 19

Which of the following series diverges by the Integral Test?

Accepted Answer

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

Half-Credit: Convergence Tests and Series in Calculus II

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