calc bc sequences and series

what does a_n denote

sequences (and infinite sequences)

what is an infinite sequence

function whose domain is a set of positive integers

when does a sequence converge

when lim a_n = L, the sequence converges to a finite number L

when does a sequence diverge

when it does not have a finite limit

infinite series of composed of what terms

a_n (sequence of real numbers) (general term of the series)

what is a p-series

(when p=1, it is called the harmonic series, which diverges)

what is a geometric series

series with first term, a, and common ratio of terms, r

when an infinite series converges, what is the limit?

when an infinite series converges, the limit is 0

if a_n does not approach 0, the series diverges

is whether a_n approaches zero sufficient to test for the convergence of a series?

no, the condition that a_n approach zero is necessary (the series cannot converge if the terms don’t approach 0), but not sufficient for the convergence of a series

ex. harmonic series

what can be done to a series without affecting its convergence/divergence?

a finite number of terms can be added/deleted

the terms of a series may be multiplied by a nonzero constant

the sums will usually differ, but convergence/divergence will remain the same

if two series are added, will the resulting series converge?

if the two series both converge, so does the resulting series

if the terms of a convergent series are regrouped, does the new series still converge?

yes, if the terms of a convergent series are regrouped, the new series converges

when does a geometric series converge

a geometric series converges if and only if |r|<1

what is the sum of a geometric series

the sum of a geometric series that converges is a/(1-r)

what is a nonnegative series

a nonnegative series is a series if a_n>=0 for all n

when does a nonnegative series converge

if f(x) is continutout, positive, decreasing function and f(n) = a_n’

improper integral from [1, infinity) f(x) dx converges

when does a p-series converge

a p-series converges if p>1, diverges if p<=1

when comparing the general terms of two series, one known to converge/diverge and one which is a nonnegative series whose convergence is unknown, when does the unknown series converge?

if the known series converges and the unknown series <= known series, the unknown series converges

if the known series diverges and the unknown series>= the known series, the unknown series diverges

*you can manipulate the terms by discarding a finite number of terms or multiplying each term by a nonzero constant

how can limits be compared to find the convergence/divergence of a series

a_n=nonnegative series, convergence is unknown

b_n=nonnegative series, convergence is known

if lim_n→infinity a_n/b_n = L

0<L<infinity

a_n’s and b_n’s convergence are the same

if lim_n→infinity a_n/b_n = 0

and b_n is convergent

a_n and b_nare both convergent

if lim_n→infinity a_n/b_n = infinity

and b_n is divergent

a_n and b_n are both divergent

if the limit of a nonnegative series exists when (a_n+1/a_n) approaches infinity, does the series converge?

if L<1, the series a_n converges

if L>1, the series diverges

if L=1, the series is inconclusive

*when used on a power series, the ratio should be written with absolute value, because it could be possible that x<0, (the ratio should be non-zero)

how do we test for the convergence of a series with a root using limits?

if the limit of (a_n)^1/n approaches infinity exists

the series converges if L<1

the series diverges if L>1

if L=1, the test is inconclusive

how can the convergence of a series, all of whose terms are negative, be tested

any test that can be applied to a nonnegative series can be used

when does an alternative series converge

when a_n+1<a_n for all n

when lim as n approaches infinity for a_n = 0

what does it mean for a series to converge absolutely or converge conditionally

absolute convergence: a series obtained by taking the absolute values of the terms of a series with mixed signs converges

Conditional convergence: when a series converges but not absolutely, ex. alternating harmonic series b/c it converges, but harmonic series diverges

what is the remainder after n terms

the error

the difference between the approximation and the true limit

denoted by R_n

|R_n|<a_n+₁

what is a power series in x

a’s are constants

if x is replaced by a real number, the power series becomes a series of constants

if x=0, series converge

what is a power series in (x-a)

a’s are constants

if x is replaced by a real number, the power series becomes a series of constants

if x=a, series converges

what is the radius of convergence

r

power series in x converges when |x|<r

power series in x diverges when |x|>r

power series in (x-a) converges when |x-a|<r

power series in (x-a) diverge when |x-a|>r

what is the interval of convergence

set of all values for x for which a power series converges

how do you find the interval of convergence

determine the radius of convergence with the Ratio Test to the series of absolute values

check each endpoint to determine whether the series converges/diverges there

what function has the domain as the interval of convergence of a power series

*continuous for each x in domain

series formed by differentiating the terms of the series converges to what

the series formed by differentiating the terms of the series converges to f’(x) for each x within the radius of convergence

*radius of convergence is the same for the power series and its derived series, but the interval of convergence may be different

what does a series obtained by integrating the terms of a given series converge to?

what is the Taylor series

series of function f, represented by a power series, about the number a

what function represented by a power series is of the Taylor Series

on an interval |x-a|<r

the coefficients are given by

c_n=(f⁽ⁿ⁾(a)) / (n!)

what are the requirements of the Taylor series

there is never more than one power series in (x-a) for f(x)

the function and all its derivatives exist at x=a

what is the Maclaurin series of a function f

the expansion of f about x=0

when a=0

what are the Maclaurin series you need to know for the AP exam?

f(x)= sin x

f(x)= cos x

f(x)= e^x

geometric series f(x)= 1/(1-x)

what are 6 functions that fail to generate a specific series in (x-a) because they and/or one or more derivatives do not exist at x=a

what is a Taylor polynomial P_n(x) of order n

the approximation of function f(x) at the point x=a

what is the relationship between Taylor polynomials and its first n derivatives

they agree at a, with f and its first n derivatives

what is the order of a Taylor polynomial

the order of the highest derivative

polynomial’s last term

what is the Maclaurin polynomial of order n that approximates f(x)

what is the tangent-line approximation to f(x) near zero

what degree does a Taylor polynomial have

a Taylor polynomial has degree n if it has powers of (x-a) up through the nth

the degree is less than n if f⁽ⁿ⁾(a)=0

what is the Lagrange remainder used for

placing an upper bound on the error of a nonnegative Taylor series

what is the Lagrange Remainder and how is it found

what is Euler’s formula

when a power series’s x is replaced by the complex number yi

e^yi= (1- (y²)/(2!) +y⁴/4! + …) + i ((y) - ((y³)/(3!)) + (y⁵)/(5!) - …)

the two series in the parenthesis converge to cos y and sin y so

e^yi=cos y + i sin y

e^{i pi} = -1