AP Precalculus- units 1-5

Arithmetic sequence: explicit

a_n=a₁+d(n-1)

Arithmatic sequence: recursive

a_n=a_n-1+d

Geometric sequence: explicit

a_n=a₁×r^n-1

arithmatic series equation (sum)

S=n/2(a₁+a_n)

Arithmatic series: sigma notation

Σ1/2^n-1=1+1/2+1/4… =2

Geometric series equation (defined)

S_n=(a₁×(1-rⁿ))/(1-r)

Geometric series equation (infinite)

S∞=a₁/(1-r)

sigma notation geometric series (infinite)

Σa_r^n-1=a/(1-r), |r|<1

convergence vs divergence: if r≥1…

diverge

convergence vs divergence: if r<1…

converge

if a series is both bounded and monotone it is

convergent

a monotone series is always

increasing or decreasing, never both

a divergent sequence goes to

infinity

a convergent sequence goes to

a number

basic divergence test

1/n^{highest degree of denomenator/numerator}

if not 0 diverge

if 0 converge or diverge

geometric series

Σ_n=1 a×r^n-1|r|<1 converge, |r|≥1 diverge

p series

Σ_n=11/n^P |P|>1 converge |P|≤1 diverge

telescopic series

Σ_a=0(1/n)-1/(n-1)

Limit comparison test

Parent function or a_n: 1/n²

Transformed function or b_n: 1/(n²-6)

divide smaller/larger

a_n/b_n=L

if a_n converges/diverges, b_n does too

Ratio test

|(a_n+1)/a_n|

<1 converges

> diverges

=1 inconclusive

direct comparison test

0≤a_n≤b_n

a_n= larger

b_n= smaller

alternating series test

Σ_n=1(-1)ⁿa_n will converge if 1)a_n=0

2) a_n+1≤a_n(sequence is decreasing)

absolute value test

if Σ|a_n| diverges but Σa_n converges, the series is conditionally divergent

if Σ|a_n| converges then Σa_n also converges. This series is absolutely convergent

if Σ|a_n| diverges and Σa_n also diverges, then the series is divergent

exponent function

number^variable

power function

variable^power

steps to graph a rational function

1) factor numerator

2) find vertical asymptote + x-int

3) find y- intercept

4) find horizontal or slant asymptotes

5) plug in points

6) domain and range

first step of graphing the rational function

factor numerator and denomenator

how to find the verticle asymptote and x intercept rational function

VADEN: what makes factors of the denomenator equal 0

XNUM: what makes numerator = 0

how to find y intercept rational function

YCONST: take constants in the original problems numerator and denomenator and divide

how to find the horizontal or slant asymptote rational functions

1) HOT- higher degree on top, use synthetic or long division without remainder to find slant asymptote

2) BOB0- bigger on the bottom =0 horizontal asymptote

3) N=D leading coefficient/leading coefficient = horizontal asymptote

steps of graphing a polynomial

1) factor (if you can)

2) find highest degree

3) find y intercept

4) find x intercept

5) find end behavior using limit notation

how to find the y intercept of a polynomial

when x=0, y=?

how to find x intercept of a polynomial

set each factor = to 0 (REMEMBER MULTIPLICITIES)

how to plot end behavior using limit notation

f(x) →∞ x→-∞ and such

concave up charactaristics

1) decreasing → increasing

2) negative→ positive

concave down characteristics

1) increase → decrease

2) positive → negative

imaginary number graphing

y axis= imaginary axis

x axis= real axis

z= x+yi

draw a vector from orgin to point