AP CALCULUS AB UNIT 4

R

all real numbers

Z

all integers

V

for all…

E

an element of…

sequence

a function whose domain is a set of non-negative numbers

recursive

elements defined by previous elements

explicit

elements defined by an explicit rule; not dependent on previous elements

if the limit of A as N appr INF equals L

- lim_{n → inf}(A_n) = L

then sequence A converges to L

if limit as A as N appr INF DNE
- lim_{n → inf}(A_n) DNE

then sequence A diverges

absolute value sequence theorem

if: limit of |A| as N appr INF equals 0
- lim_{n → inf}(|A_n|) = 0
then: limit of A as N appr INF equals 0

- lim_{n → inf}(A_n) = 0

f(x) = Cxⁿ

= C fⁿ(x) = C * n!

lim_{n → inf} (1 + (1/n))ⁿ = ?

e

series

sum of the terms of an infinite sequence

S_n

partial sum

if a sequence of partial sums (S_n) diverges

then ΣA_n is a divergent series

if a sequence of partial sums (S_n) converges

then ΣA_n is a convergent series
and
Σ_n=1^inf (A_n) = lim_{n → inf}S_n = S

telescoping series

partial sums that collapse in on themselves (“cancel”)
often in the form Σ (c/(n + d)) - (j/(n+k))

geometric form

Σ Arⁿ

divergence of geometric series

if |r| >= 1
then series diverges

convergence of a geometric series

if |r| < 1
then series converges to a/(1 - r)

test for series divergence (aka nth term test)

if lim_{n → inf} (A_n) does not equal 0
then Σ_n=1^inf (A_n) diverges

p-series

when the denominator increments but the numerator does not/remains constant

Σ_{n = 1}^inf (1/n^p) for p > 1

p-series convergence

when p > 1

p-series divergence

when 0 < p <= 1

harmonic series

a p-series in which p = 1; ALWAYS DIVERGES

Σ_n=1^inf (1/n)