AP Calculus: Power Series flashcards

Question 1

Define a power series centered at x = 0.

Accepted Answer

Sum from n=0 to infinity of c_n x^n, with powers of x.

Question 2

Define a power series centered at x = a.

Accepted Answer

Sum from n=0 to infinity of c_n (x - a)^n, with powers of (x - a).

Question 3

Expanded, what does the power series centered at 0 look like?

Accepted Answer

c_0 + c_1 x + c_2 x^2 + ...

Question 4

Expanded, what does the power series centered at a look like?

Accepted Answer

c_0 + c_1 (x - a) + c_2 (x - a)^2 + ...

Question 5

For a power series centered at a, what is the radius of convergence R if it converges only at x = a?

Accepted Answer

R = 0

Question 6

For a power series centered at a, what is the radius of convergence R if it converges for every real x?

Accepted Answer

R = infinity

Question 7

If a power series centered at a has radius of convergence R (a positive number), for which x does it converge and for which does it diverge?

Accepted Answer

It converges when |x - a| < R and diverges when |x - a| > R.

Question 8

Must a power series with a finite interval of convergence also converge at the endpoints?

Accepted Answer

Not necessarily: it may or may not.

Question 9

The geometric series sum x^n (n from 0 to infinity) equals what, and for which x?

Accepted Answer

1/(1 - x), for |x| < 1.

Question 10

How do you differentiate a power series term by term?

Accepted Answer

Differentiate each term separately; sum c_n (x - a)^n becomes sum from n=1 of n c_n (x - a)^(n-1).

Question 11

How do you integrate a power series term by term?

Accepted Answer

Integrate each term separately; the result is C + sum from n=0 of c_n (x - a)^(n+1)/(n+1).

Question 12

What is the nth Taylor polynomial for f at x = a?

Accepted Answer

p_n(x) = f(a) + f'(a)(x - a) + f''(a)/2! (x - a)^2 + ... + f^(n)(a)/n! (x - a)^n

Question 13

What are Taylor polynomials used for?

Accepted Answer

Approximating a function near a value x = a.

Question 14

What is a Maclaurin polynomial?

Accepted Answer

A Taylor polynomial centered at 0.

Question 15

What is a Maclaurin series for f?

Accepted Answer

The Taylor series for f at x = 0.

Question 16

What is the Taylor series for f at x = a?

Accepted Answer

Sum from n=0 of f^(n)(a)/n! (x - a)^n.

Question 17

How are the nth degree Taylor polynomials of f related to the Taylor series of f?

Accepted Answer

Each one is a partial sum of that Taylor series.

Question 18

When does the Taylor series of f converge to f?

Accepted Answer

If and only if the limit as n goes to infinity of the remainder R_n(x) = f(x) - p_n(x) is 0.

Question 19

For which x do the Taylor series of e^x, sin x and cos x converge to the respective functions?

Accepted Answer

For all real x.

Question 20

What is the binomial series?

Accepted Answer

The Maclaurin series for (1 + x)^r.

Question 21

What is a nonelementary integral?

Accepted Answer

An integral whose integrand has an antiderivative that cannot be written as an elementary function.

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