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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