Compute $$\lim\limits_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. If $$the limit is less than $$1$$, the series converges absolutely.

$$\lim\limits_{n \to \infty} a_n = 0$$

Question 1

Which of the following series converges by the Ratio Test?

Accepted Answer

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

Question 2

Consider the series $$\sum\limits_{n=1}^{\infty} \left( \frac{1}{2^n} - \frac{1}{3^n} \right). $$What is its sum?

Accepted Answer

$$\frac{1}{2}$$

Question 3

Which of the following tests is most appropriate to determine the convergence of $$\sum\limits_{n=2}^{\infty} \frac{1}{n \ln(n)}$$?

Accepted Answer

Integral Test

Question 4

Does the series $$\sum\limits_{n=2}^{\infty} \frac{1}{n \ln(n)}$$ converge or diverge?

Accepted Answer

It diverges by the Integral Test.

Question 5

Which of the following is the correct power series representation for $$g(x) = \frac{4}{2 - x^2$}$$ centered at $$0$$?

Accepted Answer

$$2 \sum\limits_{n=0}^{\infty} \left( \frac{x^2$}{2} \right)^n$$

Question 6

What is the interval of convergence for the power series $$\sum\limits_{n=0}^{\infty} \left( \frac{x^2$}{2} \right)^n$$?

Accepted Answer

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

Question 7

Find the Taylor series for $$f(x) = \sin(x)$$ about $$x = 0.$$

Accepted Answer

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

Question 8

Which of the following series is absolutely convergent?

Accepted Answer

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

Question 9

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

Accepted Answer

Compute $$\lim\limits_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight|. If $$the limit is less than $$1$$, the series converges absolutely.

Question 10

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

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \frac{n^2$}{4n^2$ + 1}$$

Question 11

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

Accepted Answer

Check if the terms decrease in absolute value and approach zero.

Question 12

Which of the following is a necessary condition for the convergence of $$\sum\limits_{n=1}^{\infty} a_n$$?

Accepted Answer

$$\lim\limits_{n 	o \infty} a_n = 0$$

Question 13

Which of the following series converges conditionally but not absolutely?

Accepted Answer

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

Question 14

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

Accepted Answer

Integrate $$f(x)$$ from $$1 to$$ $$\infty$$ and check if the integral converges.

Question 15

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

Accepted Answer

If $$\sum\limits_{n=1}^{\infty} \frac{1}{n^p}$$, the series converges if $$p \u003e 1$$ and diverges if $$p \leq 1.$$

Question 16

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

Accepted Answer

Compute $$\lim\limits_{n 	o \infty} \sqrt[n]{|a_n|}. If $$the limit is less than $$1$$, the series converges absolutely.

Question 17

Which of the following is the correct process for the Direct Comparison Test?

Accepted Answer

Compare $$a_n to$$ $$b_n$$ term-by-term, where $$0 \leq a_n \leq b_n$$ for all $$n.$$

Question 18

Which of the following is the correct process for the Limit Comparison Test?

Accepted Answer

Compute $$\lim\limits_{n 	o \infty} \frac{a_n}{b_n} = L$$, where $$0 \u003c L \u003c \infty.$$

Convergence of Series and Power Series in Calculus II

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