series + sequences

monotonic sequence

always increasing or decreasing

bounded sequences

upper and lower bound (if both are true then then a_n is bounded)

infinite series

n=1 SUM infinity a_n = a1 + a2 +a3 … an
a_n gives the value of the nth term in the sequence

partial sum

S_n = a1 + a2 + a3 …

If partial sum converges then the infinite series converges
Value the infinite series converges to is equal to lim n→ infinity of S_n

geometric series converges to

(ar^k) / (1-r) where (ar^k) is the first term in the series

geometric series reminder

be careful of (-n) powers because it can flip R (common ratio)

geometric series written as

• a_orⁿ

• a₁r^n-1

• a₂r^n-2

nth term test

• Lim n→ infinity = 0 because series could converge or diverge

• if it doesnt equal zero then it diverges

  • REVERSE ISNT TRUE THOUGH; IF IT DIVERGES YOU CAN’T ASSUME WHAT THE LIMIT IS

• sometimes you have to use l’hopitals to evaulate limit

integral test CONDITIONS

ALWAYS CHECK FOR: positive, continuous, decreasing

“limit of the series” means just take the limit of a_n

if limit is finite then BOTH CONVERGE if not BOTH DIVERGE

dont forgettt

you can put them into your calculator

partial sums using: math 0

DONT FORGET NOTATION

watch the integral bounds and the limit for the improper integral

limit comparison test conditions

BOTH A_N AND B_N MUST BE POSITIVE

limit comparison test

BOTH converge or BOTH diverge

absolute/conditional convergence

if abs value of a_n converges than a_n also converges (absolute); if the abs value version diverges and the normal converges its conditional; if both diverge its divergent

when solving for interval of convergence make sure to

test the endpoints

alternating series error bound

error is less than the 1st term left off (aka the next term)

lagrange error bound

MAX f ⁽ⁿ⁺¹⁾(z)(x-c)ⁿ⁺¹ / (n+1)!

radius/int of convergence

use ratio test; (dont forget abs value)
if its <1 then it converges to an interval
= 0 converges R (-infinity, infinity) — all real numbers

>1 converges to center x=c

Taylor/Maclaurin Series

applies for f(x) if it’s differentiable for every order

e^x

1 + x + x²/2! + x³/3! … sum x^n / n!

sinx

x - x³ / 3! + x^5 / 5! + … sum (-1)ⁿx²ⁿ⁺¹ / (2n+1)!

cosx

1 - x² / 2! + x^4 / 4! + … sum (-1)ⁿx²ⁿ / (2n)!

1 / (1+x)

1 - x + x² - x³ + … + sum (-1)² xⁿ