It converges because $$\lim_{n \to \infty} a_n = 0.$$

Question 1

Which of the following is the correct Maclaurin series for the function $$f(x) = x \cos(x)$$?

Accepted Answer

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

Question 2

Given the sequence $$a_n = \frac{1}{n}$$, determine whether it converges or diverges.

Accepted Answer

It converges because $$\lim_{n 	o \infty} a_n = 0.$$

Question 3

Consider the series $$\sum_{n=1}^{\infty} \frac{1}{2^n}. Is it $$geometric, telescoping, or neither? What is its sum?

Accepted Answer

It is geometric with sum $$1.$$

Question 4

Given the series $$\sum_{n=1}^{\infty} \frac{1}{n^2$}$$, which test can be used to determine convergence, and what is the result?

Accepted Answer

The integral test shows convergence.

Question 5

Suppose you want to estimate $$\sin(5)$$ using its Maclaurin series up to the fourth term. Which expression would you use?

Accepted Answer

$$5 - \frac{5^3$}{3!} + \frac{5^5$}{5!} - \frac{5^7$}{7!}$$

Question 6

Write the Maclaurin polynomial to the fifth term for $$f(x) = x \cos(x). $$Which is correct?

Accepted Answer

$$x - \frac{x^3$}{2} + \frac{x^5$}{24}$$

Question 7

For the sequence $$a_n = n^2$, use the difference test to determine if it is strictly increasing, strictly decreasing, or neither.

Accepted Answer

Strictly increasing

Question 8

Prove that $$e^{i\pi} = -1$ using the Maclaurin series for e^x$. Which step is correct?

Accepted Answer

Substitute $$x = i\pi$$ into $$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$ and simplify.

Question 9

Integrate $$f(x) = x \cos(x)$$ using its Maclaurin series. Which is the correct approach?

Accepted Answer

Integrate term by term after expanding $$x \cos(x) as a $$Maclaurin series.

Question 10

Integrate $$f(x) = \ln(|x|)$$ from $$-1 to$$ $$0. $$What is the result?

Accepted Answer

The integral does not exist because $$\ln(|x|) is $$undefined at $$x=0.$$

Question 11

Write the Taylor series for $$f(x) = \ln(x) at$$ $$x_0 = 1$. Which is correct?

Accepted Answer

$$\sum_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$}{n} (x-1)^n$$

Question 12

Given the series $$\sum_{n=1}^{\infty} \frac{1}{n}$$, which test can be used to determine convergence, and what is the result?

Accepted Answer

The integral test shows divergence.

Question 13

For the sequence $$a_n = \frac{n}{n+1}$$, use the derivative test to determine its behavior.

Accepted Answer

Strictly increasing

Question 14

Find the radius of convergence for the Maclaurin series of $$f(x) = x \cos(x).$$

Accepted Answer

Infinity

Question 15

Which of the following is the general term for the sequence $$a_n = \frac{1}{n^2$}$$?

Accepted Answer

$$a_n = \frac{1}{n^2$}$$

Question 16

Which of the following is a real-life application of the Maclaurin series for $$\sin(x)$$?

Accepted Answer

Approximating the motion of a pendulum for small angles.

Question 17

Which test is most appropriate to determine the convergence of $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ for $$p \u003e 1$$?

Accepted Answer

Integral test

Question 18

If you use the ratio test on the series $$\sum_{n=1}^{\infty} \frac{n!}{2^n}$$, what is the result?

Accepted Answer

The series diverges.

Question 19

Which of the following is the sum of the telescoping series $$\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right)$$?

Accepted Answer

1

Calculus II Series and Sequences Review – Step-by-Step Study Guidance

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