Taylor CHAPTER and Maclaurin 2 Series Theorem If

Taylor. CHAPTER and Maclaurin 2 Series Theorem: If f 2. 4 has a Continuity power series representation (expansion) at a, that is, if f(x) = n=0 cn (x – a)n |x – a| < R Then its coefficients are given by the formula cn = ( f (n)(a) ) / n!. f(x) = n=0 [ ( f (n)(a) ) / n! ] (x – a)n = f (a) + [ f’(a) / 1!] (x – a) + [ f’’(a) / 2!] (x – a)2 + … f(x) = n=0 [ ( f (n)) (0) ) / n! ] (x – a)n = f (0) + [ f’(0) / 1!] x + [ f’’(0) / 2!] x 2 + …

CHAPTER 2 Theorem: If f (x) = Tn (x) + Rn (x), where Tn is the nth-degree Taylor polynomial of f at a and 2. 4 Continuity lim n= Rn (x) = 0 for | x – a | < R, then f is equal to the sum of its Taylor series on the interval | x – a | < R. Taylor’s Inequality: If | f (n+1)(x)| M for | x – a | < R, then the remainder Rn (x) of the Taylor series satisfies the inequality | Rn (x) | M | x – a | n+ 1 / (n + 1)! for | x – a | < R.

CHAPTER 2 lim n-> x n / n ! = 0 for every real number x. e x = n=0 2. 4 x n / n. Continuity ! for all x. e = n=0 1/n ! = 1 + 1 / 1! + 1 / 2! + 1 / 3! +… sin x = x – x 3 / 3! + x 5 / 5! + x 7 / 7! + … = n=0 (-1) n ( x 2 n+ 1 / (2 n + 1)! ) for all x. cos x = 1 – x 2 / 2! + x 4 / 4! + x 6 / 6! + … = n=0 (-1)n x 2 n / (2 n)! for all x.