maclaurin and taylor walk into a bar and have a series of shots + error bounds

taylor series centered at x = a

f(x) ~~ P_n(x) = f(a) + f’(a)(x-a) + f”(a)/2!(x-a)² + f’’’(a)/3! (x-a)³ + …

maclaurin series of 1/1-x

1 + x + x² + x³ + … + x^n + …

maclaurin series of 1/1+x

1 - x + x² - x³ +… (-1)^n x^n + …

maclaurin series of e^x

1 + x + x²/2! + x³/3! + … + x^n/n! + …

maclaurin series of cos x

1 - x²/2! + x^4/4! - x^6/6! + x^8/8! + … + series n = 0 to infinity (-1)^n x^ 2n / (2n)!

maclaurin series of sin x

x - x³/3! + x^5/5! - x^7/7! + … + series n = 0 to infinity (-1)^n x^ 2n-1 / (2n-1)!

alternating series bound

|error| ≤ |S_n+1 - S_n|, can be used with convergent series

lagrange error bound

upper limit on error of taylor series approximation; |f(c) - T (c)| ≤ M|c - a|^(n+1) / (n+1)! - a is the center, n is the degree of taylor series, M = f⁽ⁿ⁺¹⁾(a)