Series & Infinite Sequences Formulas, etc

AP Calc BC series formulas, tests, r rules, etc In progress

monotonic

terms are nonincreasing or nondecreasing (can be 0)

checking for monotonicity

taking the derivative (f’(n) ≤ or ≥ 0), proving using nth term (n+1 ≤ or ≥ n)

bounded sequence

sequence bounded above and below

limit of a sequence

if An is a sequence such that f(n) = An for every positive integer, lim n -> infinity = lim n -> infinity f(n) = L where L is a real number

let lim n -> infinity An = L, lim n -> infinity Bn = K
addition property of limits of sequences

lim n -> infinity (An +/_ Bn) = L + K

let lim n -> infinity An = L

C is a real number
Scalar property of limits of sequences

lim n -> infinity CAn = CL

let lim n -> infinity An = L, lim n -> infinity Bn = K

Multiplicative property of limits of sequences

lim n -> infinity (AnBn) = LK

let lim n -> infinity An = L, lim n -> infinity Bn = K

Bn & K are not equal to 0

Division property of limits of sequences

lim n -> infinity (An/Bn) = L/K

absolute value theorem of limits of sequences

if lim n -> infinity |An| = 0, lim n -> infinity An = 0

Bounded monotonic sequences theorem

if a sequence is bounded and monotonic, it converges

Infinite series convergence/divergence

An infinite series converges if the sequence of it’s partial sums converges, otherwise it diverges

Geometric series form

Σn=0 to infinity ar^n

Geometric series converge when

|r| < 1

Geometric series diverge when

|r| ≥ 1

When a geometric series converges, the sum is given by

Σn=0 to infinity a(or first term)/1-r

in some cases, if Σn=0 to infinity converges

then lim n→infinity An = 0

Nth term test

if lim n→infinity An is not equal to 0, the sum diverges

Harmonic series

Σn=1 to infinity 1/n, diverges

P-series test form

Σn=1 to infinity 1/n^p

P-series converges when

p > 1

P-series diverges when

0 < p ≤ 1

Telescoping series

Collapses/converges, commonly Σ(1/n)-1/(n+1)

Direct comparison test

0 ≤ An ≤ Bn

DTC convergence

If ΣBn converges, ΣAn converges

DTC divergence

If ΣAn diverges, then ΣBn diverges

Limit comparison test

If An and Bn > 0 and lim n → infinity (An/Bn) = L, and L is finite and positive, both series either diverge or converge

Alternating series test

An > 0

Σ n=1 to infinity (-1)^n An & Σ n=1 to infinity (-1)^(n+1) An converge if

1. lim n→infinity An = 0

2. A(n+1) < An for all n (the |terms| is decreasing)

Alternating Series Test - Absolute convergence

Σ |An| converges

An Alternating Series - Conditional Convergence

Σ An converges, Σ |An| diverges

Alternating series error

For an alternating series, the difference between the actual sum of the series and the sum of the first N terms will always be less than or equal to the next neglected term

lim n→infinity of (1+k/n)^n

e^k

lim n→infinity of an/√(bn²+c)

a/√b

lim n→infinity of the nth root of a

1

lim n→infinity of the nth root of n

1

(n+1)!/n!

n+1

Ratio test Convergence

lim n→infinity of |A(n+1)/An| < 1

Ratio test Divergence

lim n→infinity of |A(n+1)/An| > 1

Ratio test failure

lim n→infinity of |A(n+1)/An| = 1

Root test convergence

lim n→infinity of the nth root of √|An| < 1

Root test divergence

lim n→infinity of the nth root of √|An| > 1

Root test failure

lim n→infinity of the nth root of √|An| = 1