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Taylor and Maclaurin Series

Taylor series of f at a:

Taylor and Maclaurin Series

( )

f

(a)

f (a)

f (x) =

(x

a) = f (a) +

(x

a)

n!

1!

=0

Per-Olof Persson

f (a)

f (a)

2

3

+

(x

a)

+

(x

a)

+

2!

3!

Maclaurin series if a = 0:

Department of Mathematics

University of California, Berkeley

( )

f

(0)

f (0)

f (x) =

x = f (a) +

x

Math 1B Calculus

n!

1!

=0

f (0)

f (0)

2

3

+

x

+

x

+

2!

3!

Remainders

Common Maclaurin Series

Theorem

1

2

3

=

x = 1 + x + x

+ x

+

R = 1

If f (x) = T (x) + R (x), where T is nth partial sum of the

1

x

=0

Taylor polynomial of f at a, and

2

3

x

x

x

x

e =

= 1 +

+

+

+

R =

n!

1!

2!

3!

lim R (x) = 0

=0

2 +1

3

5

x

x

x

sin x =

( 1)

= x

+

R =

for x

a < R, then f = sum of its Taylor series on x

a < R

(2n + 1)!

3!

5!

=0

2

2

4

x

x

x

Theorem (Taylor’s Inequality)

cos x =

( 1)

= 1

+

R =

(2n)!

2!

4!

=0

( +1)

If f

(x)

M for x

a

d, then the remainder R (x) of

2 +1

3

5

x

x

x

1

the Taylor series satisﬁes

tan

x =

( 1)

= x

+

R = 1

2n + 1

3

5

=0

M

+1

k

k(k

1)

R (x)

x

a

for x

a

d

2

(1 + x) =

x = 1 + kx +

x

+

R = 1

(n + 1)!

n

2!

=0

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