Question 1

Which of the following is the general form of a power series centered at $$x = a$$?

Accepted Answer

$$\sum_{n=0}^{\infty} c_n (x - a)^n$$

Question 2

According to Theorem 18, if $$\sum_{n=0}^{\infty} a_n x^n$$ converges at $$x = c 
eq 0$$, for which values of $$x$$ does the series converge absolutely?

Accepted Answer

$$|x| \u003c |c|$$

Question 3

What is the radius of convergence $$R$$ for the power series $$\sum_{n=0}^{\infty} c_n (x - a)^n if $$the series converges for all $$x$$?

Accepted Answer

$$R = \infty$$

Question 4

Given the power series $$\sum_{n=0}^{\infty} \frac{x^n}{2^n}$$, what is its interval of convergence?

Accepted Answer

$$(-2, 2)$$

Question 5

If $$\sum_{n=0}^{\infty} a_n x^n$$ diverges at $$x = d$$, what can be concluded about the series for $$|x| \u003e |d|$$?

Accepted Answer

The series diverges for $$|x| \u003e |d|$$

Question 6

Which of the following is the Maclaurin series for $$\sin x$$?

Accepted Answer

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

Question 7

Suppose $$f(x) = e^x. $$What is the Taylor polynomial of order 2 centered at $$x = 0$$?

Accepted Answer

$$1 + x + \frac{x^2$}{2}$$

Question 8

If $$A(x) = \sum_{n=0}^{\infty} a_n x^n$$ and $$B(x) = \sum_{n=0}^{\infty} b_n x^n$$ both converge absolutely for $$|x| \u003c R$$, what is the coefficient $$c_2 in $$the product $$A(x)B(x)$$?

Accepted Answer

$$a_2$$ $$b_0$$ + $$a_1$$ $$b_1$$ + $$a_0$$ $$b_2$$

Question 9

Which theorem guarantees that the derivative of a power series can be found by differentiating term by term within the radius of convergence?

Accepted Answer

Theorem 21

Question 10

Given $$f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n$$ with radius of convergence $$R$$, what is the interval of convergence for $$\int f(x) dx$$?

Accepted Answer

$$a - R \u003c x \u003c a + R$$

Question 11

Which of the following is the Maclaurin series for $$\ln(1 + x)$$?

Accepted Answer

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

Question 12

If $$\sum_{n=0}^{\infty} a_n x^n$$ converges for $$|x| \u003c 2$$ and $$f(x) is $$continuous, for which $$x$$ does $$\sum_{n=0}^{\infty} a_n (f(x))^n$$ converge absolutely?

Accepted Answer

$$|f(x)| \u003c 2$$

Question 13

Which of the following statements about the interval of convergence for a power series $$\sum_{n=0}^{\infty} c_n (x - a)^n is $$correct?

Accepted Answer

The series may converge at one, both, or neither endpoint.

Question 14

Find the first three nonzero terms of the Maclaurin series for $$e^x \sin x.$$

Accepted Answer

$$x + x^2$ + \frac{1}{3}x^3$

Question 15

Which of the following is the correct Taylor series for $$f(x) = \cos x$$ centered at $$x = 0$$?

Accepted Answer

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

Question 16

Suppose $$\sum_{n=0}^{\infty} c_n (x - a)^n$$ has radius of convergence $$R. $$What is the radius of convergence for its term-by-term derivative?

Accepted Answer

$$R$$

Question 17

If $$f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n$$ and $$f$$ has derivatives of all orders, what is the general formula for the Taylor coefficients $$c_n$$?

Accepted Answer

$$c_n = \frac{f$$^{(n)}$$(a)}{n!}$$

Power Series and Taylor Series: Definitions, Convergence, and Operations

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