Question 1

Which of the following is the general form of the $$nth $$order Taylor polynomial for a function $$f(x)$$ centered at $$a$$?

Accepted Answer

$$p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2$ + \cdots + \frac{f$$^{(n)}$$(a)}{n!}(x-a)^n$$

Question 2

Suppose you want to approximate $$\sin(0.1) to $$within $$10^{-30}$$ using its Taylor polynomial centered at $$0. $$What is the smallest $$n$$ such that the error is guaranteed to be less than $$10^{-30}$$?

Accepted Answer

The smallest $$n$$ such that $$\frac{(0.1$$)^{n+1}$$}{(n+1)!} \leq 10$$^{-30}$$

Question 3

Given the power series $$\sum_{k=0}^{\infty} a_k (x-a)^k$$, what is the definition of the radius of convergence $$R$$?

Accepted Answer

The distance from the center $$a to $$the boundary of the interval where the series converges absolutely.

Question 4

For the power series $$\sum_{k=0}^{\infty} \left(\frac{x}{10}\$$right)^{2k}$$, what is the interval of convergence?

Accepted Answer

$$(-10, 10)$$

Question 5

Which test is commonly used to determine the radius of convergence for a power series?

Accepted Answer

Ratio Test or Root Test

Question 6

If a power series $$\sum_{k=0}^{\infty} a_k (x-a)^k$$ has radius of convergence $$R=\infty$$, what is its interval of convergence?

Accepted Answer

$$(-\infty, \infty)$$

Question 7

Given $$f(x) = \frac{1}{1-x}$$, what is its power series representation centered at $$0$$?

Accepted Answer

$$\sum_{k=0}^{\infty} x^k$$

Question 8

What is the interval of convergence for the power series $$\sum_{k=0}^{\infty} x^k$$?

Accepted Answer

$$(-1, 1)$$

Question 9

Suppose $$f(x)$$ has a power series representation with radius of convergence $$R. $$What happens to the radius of convergence when you differentiate or integrate the series term by term?

Accepted Answer

The radius of convergence remains unchanged.

Question 10

Given $$f(x) = \frac{1}{1-x}$$, what is the power series representation for $$\ln(1-x)$$ centered at $$0$$?

Accepted Answer

$$-\sum_{k=1}^{\infty} \frac{x^k}{k}$$

Question 11

For the power series $$\sum_{k=1}^{\infty} \frac{-x^k}{k}$$, what is the interval of convergence?

Accepted Answer

$$(-1, 1)$$

Question 12

If $$f(x) = \frac{1}{1-x}$$, what is the power series representation for $$g(x) = \frac{1}{(1-x)^2$}$$?

Accepted Answer

$$\sum_{k=0}^{\infty} (k+1)x^k$$

Question 13

Which of the following statements about the interval of convergence for a power series is TRUE?

Accepted Answer

You must always check the endpoints separately after using the ratio/root test.

Question 14

Suppose $$\sum_{k=0}^{\infty} a_k x^k$$ converges for $$|x| \u003c R. $$What happens if you plug in $$x = R$$?

Accepted Answer

You must test convergence at $$x = R$$ separately; the series may or may not converge.

Question 15

Given $$f(x) = \frac{1}{1-x}$$, what is the power series representation for $$g(x) = \ln(1+x^2$)$$ centered at $$0$$?

Accepted Answer

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

Question 16

Which of the following is a valid power series representation for $$f(x) = \ln(\sqrt{1-x})$$ centered at $$0$$?

Accepted Answer

$$\sum_{k=1}^{\infty} \frac{-x^k}{2k}$$

Question 17

If $$f(x)$$ and $$g(x)$$ both have power series representations on an interval $$I$$, which of the following is TRUE about their sum $$f(x) + g(x)$$?

Accepted Answer

The sum $$f(x) + g(x)$$ has a power series representation on $$I.$$

Question 18

What is the function represented by the series $$\sum_{k=1}^{\infty} \frac{-x^k}{k}$$?

Accepted Answer

$$\ln(1-x)$$

Question 19

Which of the following is the correct power series representation for $$f(x) = \frac{1}{1+x^2$}$$ centered at $$0$$?

Accepted Answer

$$\sum_{k=0}^{\infty} (-1)^k x$$^{2k}$$

Approximating Functions with Polynomials and Power Series: Theory, Properties, and Applications

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