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**Unformatted text preview: **ntered at x = 0 for g (x)
is
∞
xn
n!
n=0
which is exactly the same Taylor Series as for ex . However at x = 2, g (2) = e while
∞ n=0 2n
= e2 = g (2).
n! Example.
Find the Taylor series centered at x = 0 for f (x) = cos(x).
We have that
f (x)
f (x)
f (x)
f (4) (x)
f (5) (x)
f (6) (x)
f (7) (x)
f (8) (x) = sin(x)
= − cos(x)
= sin(x)
= cos(x)
= − sin(x)
= − cos(x)
= sin(x)
= cos(x) DE Math 128 =⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
.
.
. f (0)
f (0)
f (0)
f (4) (0)
f (5) (0)
f (6) (0)
f (7) (0)
f (8) (0) = − sin(0)
= − cos(0)
= sin(0)
= cos(0)
= − sin(0)
= − cos(0)
= sin(0)
= cos(0) 311 =0
= −1
=0
=1
=0
= −1
=0
=1 (B. Forrest)2 4.5. Introduction to Taylor Series CHAPTER 4. Sequences and Series with the cycle repeating itself every four derivatives. This gives
f (4k) (x)
f (4k+1) (x)
f (4k+2) (x)
f (4k+3) (x) = cos(x)
= − sin(x)
= − cos(x)
= sin(x) f (4k) (0)
f (4k+1) (0)
f (4k+2) (0)
f (4k+3) (0) =⇒
=⇒
=⇒
=⇒
.
.
. = cos(0)
= − sin(0)
= − cos(0)
= sin(0) =1
=0
= −1...

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