Calculus: Power Series flashcards

Question 1

What is a power series centered at x = a?

Accepted Answer

A series of the form Σ cₙ(x − a)ⁿ, n from 0

Question 2

What is the interval of convergence of a power series?

Accepted Answer

The set of all real x at which the series converges

Question 3

If a power series centered at a converges only at x = a, what is its radius of convergence?

Accepted Answer

R = 0

Question 4

If a power series converges for all real x, what is its radius of convergence?

Accepted Answer

R = ∞

Question 5

What does a radius of convergence R > 0 mean for a series centered at a?

Accepted Answer

Convergence when |x − a| < R, divergence when |x − a| > R

Question 6

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

Accepted Answer

No: it may or may not

Question 7

Which convergence test is often used to find a radius of convergence?

Accepted Answer

The ratio test

Question 8

For |x| < 1, Σ xⁿ (n from 0) = ?

Accepted Answer

1/(1 − x)

Question 9

Term-by-term derivative of Σ cₙ(x − a)ⁿ?

Accepted Answer

Σ n·cₙ(x − a)ⁿ⁻¹, n from 1

Question 10

Term-by-term integral of Σ cₙ(x − a)ⁿ?

Accepted Answer

C + Σ cₙ(x − a)ⁿ⁺¹/(n + 1), n from 0

Question 11

If a power series converges to f on (−R, R), what does its term-by-term derivative converge to there?

Accepted Answer

f′

Question 12

Taylor series of f at x = a?

Accepted Answer

Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ, n from 0

Question 13

nth Taylor polynomial of f at x = a?

Accepted Answer

pₙ(x) = f(a) + f′(a)(x − a) + f″(a)/2! (x − a)² + ⋯ + f⁽ⁿ⁾(a)/n! (x − a)ⁿ

Question 14

What is a Maclaurin series?

Accepted Answer

A Taylor series for f at x = 0

Question 15

What is a Maclaurin polynomial?

Accepted Answer

A Taylor polynomial whose center is 0

Question 16

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

Accepted Answer

They are its partial sums

Question 17

If f can be written as a power series about x = a, what is that series?

Accepted Answer

Its Taylor series at x = a

Question 18

Taylor's theorem: what is the remainder Rₙ(x) = f(x) − pₙ(x) equal to?

Accepted Answer

f⁽ⁿ⁺¹⁾(c)/(n + 1)! · (x − a)ⁿ⁺¹ for some c between x and a

Question 19

If |f⁽ⁿ⁺¹⁾(x)| ≤ M on an interval I containing a, what bound holds for |Rₙ(x)| on I?

Accepted Answer

|Rₙ(x)| ≤ M/(n + 1)! · |x − a|ⁿ⁺¹

Question 20

For which real x do the Taylor series of eˣ, sin x and cos x converge to those functions?

Accepted Answer

For all real x

Question 21

The binomial series is the Maclaurin series of which function?

Accepted Answer

f(x) = (1 + x)ʳ

Question 22

For which x does the binomial series converge?

Accepted Answer

For |x| < 1

Question 23

First three terms of the binomial series for (1 + x)ʳ?

Accepted Answer

1 + rx + r(r − 1)/2! · x²

Question 24

What is a nonelementary integral?

Accepted Answer

An integral whose integrand's antiderivative cannot be written as an elementary function

Calculus: Power Series flashcards

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