8.9 Convergence

8.9 Convergence - 8 .9 
C o n v e r g e n c e 
o f
T...

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Unformatted text preview: 8 .9 
C o n v e r g e n c e 
o f
T a y l o r 
S e r ie s 
 Y o u 
 s h o u l d 
 k n o w 
 t h e 
 M a c l a u r in 
 s e r ie s :
 ∞ un eu = ∑ n = 0 n! u 2 n +1 sin u = ∑ (−1) (2 n + 1)! n =0 ∞ ∞ n u 2n cos u = ∑ (−1) (2 n)! n =0 n 
 
 W e 
c a n 
u s e 
t h e s e 
t o 
fin d 
s e r ie s 
fo r 
m o re 
c o m p l e x 
 fu n c t io n s .
 E x .
F in d 
t h e 
M a c l a u r in 
s e r ie s 
fo r 
s in ( ‐ 5 x ) 
 
 
 
 
 
 
 
 
 E x .
F in d 
t h e 
M a c l a u r in 
s e r ie s 
fo r 
 xe x3 .
 
 
 
 
 
 
 
 
 
 Do.
Find
the
Maclaurin
series
for
 cos 
 
 
 
 
 
 
 
 x .
 ∞ a ar if r < 1 .
 R e c a l l 
t h a t 
 ∑ 
h a s
su m 
 1 − r n =0 n W e 
c a n 
u s e 
t h is 
t o 
a p p r o x im a t e 
fu n c t io n s 
a s 
s e r ie s .
 1 E x .

 f ( x ) = 1 − x 
 
 
 
 
 
 
 1 f ( x) = E x .

 1+ x3 
 
 
 
 
 
 
 
 x2 E x .

 f ( x ) = 1 − x 3 
 
 
 
 
 
 
 ∫ .5 E x .

U s e 
s e r ie s 
t o 
e v a l u a t e 
 0 o f
m a g n it u d e 
l e s s 
t h a n 
.0 0 0 1 .
 
 
 
 
 
 
 
 x2 3 dx 
with
an
error
 1+ x E x .
E v a l u a t e 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ∫ 0.1 0 e −x 2 dx 
 ex − e−x Ex.
 lim x→ 0 x
 
 
 
 
 
 
 
 
 E x .

T h e 
fo l l o w in g 
is 
a 
M a c l a u r in 
s e r ie s 
fo r 
a 
fu n c t io n 
 f( x ) 
a t 
a 
p a r t ic u l a r 
p o in t .

W h a t 
is 
t h e 
fu n c t io n 
a n d 
w h a t 
 is 
t h e 
p o in t ? 

F in d 
t h e 
s u m 
o f
t h e 
s e r ie s .
 ∞ 3n ∑0 5n n! 
 n= 
 
 
 
 
 (−1) π 2n +1 π π3 π5 − + − ... + 2 n +1 Ex.
 4 4 3 3! 4 55! 4 (2 n + 1)! 
 n 
 
 
 
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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