1. Taylor and Maclaurin Series
CHAPTER 2
2.4 Continuity
Theorem: If f has a 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!] x2 + …

2. CHAPTER 2
Theorem: If f (x) = Tn (x) + Rn (x), where Tn is the nth-degree Taylor polynomial of f at a and
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.
2.4 Continuity
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.

3. lim n-> x n / n ! = 0 for every real number x.
CHAPTER 2
lim n-> x n / n ! = 0 for every real number x.
2.4 Continuity
e x =  n=0 x n / n ! for all x.
e =  n=0 1/n ! = / 1! + 1 / 2! + 1 / 3! +…
sin x = x – x3 / 3! + x5 / 5! + x7 / 7! + … =  n=0 (-1) n ( x2n+ 1 / (2n + 1)! ) for all x.
cos x = 1 – x2 / 2! + x4 / 4! + x6 / 6! + … =  n=0 (-1)n x2n / (2n)! for all x.

4. CHAPTER 2
1 / (1 – x ) =  n=0 xn = 1 + x + x2 + x3 + … ( -1 ,1 )
e x =  n=0 x n / n ! = 1 + x + x2 / 2! + x3/ 3!+ … (-  , )
sin x =  n=0 (-1) n x 2n+ 1 / (2n + 1)! =
x – x3/ 3! + x5 / 5! + x7 / 7! + … (-  , )
cos x =  n=0 (-1) n x 2n / (2n)! =
1 – x2 / 2! + x4 / 4! + x6 / 6! + … (-  , )
tan x =  n=0 (-1) n x2n+ 1 / (2n + 1) =
x – x3/ 3! + x5 / 5!+ x7 / 7! + … [- 1 , 1]
2.4 Continuity