PDARLING Test

AP Calculus BC

P-series test equation

1/n^p

In p-series, if p>1…

the series converges

In p-series, if 0<p=<1…

the series diverges

harmonic

1/n (diverges)

to use the p-series test, what must occur?

the degree of the numerator’s variable must be less than that of the denominator’s variable

In direct comparison test, if the bigger series converges…

the smaller series also converges

In direct comparison test, if the smaller series diverges…

the bigger series also diverges

In the alternate series test, what should you do with the alternator?

ignore

potential alternator #1

(-1)^n

potential alternator #2

(-1)^n+1

In alternating series test, what two things must be proven true for the series to be convergent

1. lim (n→infinity) of a(n) = 0

2. a(n+1) =< a(n) for all values of n

In alternating series test, if lim (n→infinity) of a(n) = 0 AND a(n+1) =< a(n) for all values of n… the series is:

convergent (with an error of a(n+1) )

A series is conditionally convergent if…

lim (n→infinity) of a(n) = 0 AND a(n+1) =< a(n) for all values of n, but the series |a(n)| diverges

A series is absolutely convergent if…

lim (n→infinity) of a(n) = 0 AND a(n+1) =< a(n) for all values of n, AND the series |a(n)| converges

Ratio test equation

L=lim(n→infinity) | (a(n+1)) / (a(n)) |

In ratio test, if L>1…

the series diverges

In ratio test, if L = 1…

Inconclusive

In ratio test, if L<1…

the series converges

Factorials (… !)

ratio test

n is in the exponent and it isn’t obviously geometric

ratio test

Limit comparison test equation

lim (n→infinity) ( a(n) ) / ( b(n) )

in limit comparison test, if the limit (p) is finite and p > 0…

the two series either both converge or both diverge (need another test to determine which the two series do)

in limit comparison test, if the limit (p) is not finite OR p =< 0…

inconclusiiv

-series is positive

-series is decreasing

-series is continuous

integral test

requirements for integral test

1. series is positive

2. series is decreasing

3. series is continuous

if the integral of the series converges to a specific number…

the series converges to a number greater than C

if the integral of the series diverges (goes to +infinity or -infinity)…

the series diverges

Nth term test equation

lim (n→infinity) a(n)

in Nth term test, if lim (n→infinity) a(n) = 0…

inconclusive, need more testing

in Nth term test, if lim (n→infinity) a(n) = anything but 0 (this includes +infinity or -infinity)…

series diverges

the Nth term test can only prove…

divergence

if completing the alternate series test and lim (n→infinity) a(n) does not equal 0 (aka, the first pre-req fails), divergence is proven via which test?

Nth term test

geometric test equation

ar^n

geometric test convergence equation

a/(1-r)

in geometric test, if |r| < 1…

series converges (to a/(1-r) )

in geometric test, if |r| >= 1

series diverges

number base raised to a variable exponent

geometric test

Multiplier present in the list of numbers in an expanded series

geometric test

variable base to a negative number exponent

p-series test

variable base in the denominator raised to a number exponent (assuming degree of the numerator’s variable is less)

p-series test

opposite format of p-series test

geometric test

opposite format of geometric test

p-series test

e

lim (n→ infinity) ( 1 + (1/n) )^n

a harmonic series automatically…

diverges

order of growth power

1. exponentials

2. polynomials (in order of degree)

3. logarithmics

in alternate series test, if limit test is passed but a(n+1) is not =< a(n) (aka the second test fails)…

the alternate series test is inconclusive

alternating harmonic equation

((-1)^n) / n OR ((-1)^n+1) / n (aka, an alternator over n)

an alternating harmonic automatically…

converges