can beCalculus
Lesson
57
Taylor
Series
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EVERY function with derivatives for all orders at can be
expressed as a Taylor Series (note: Taylor Polynomial does
note have an infinite number of terms).
We used the geometric series and integrating term-by-term in the
last section to generate power series. Now we will use the
Taylor method.
centered at 
.
If it is centered at 
then
it is a Maclauren Series.
Example: Find the Maclauren Series for 
Note that we only evaluated the series at 
, so this series is only good
near zero! What happens away from zero is anyone's guess!
example: 
Instead of worrying about the Radius of Convergence for Taylor Series, we just
always know that it converges as long as 
(we know this
from our definition of remainder of the Taylor Polynomials!) Note that
converges for all x
using the above definition.
Other tricks:
Substitution: Since we already know the series for 
, it is easy
to find the series for 
.
Also note that this series will have
the same radius of convergence as 
since we are comparing
and 
and they both go to
ONE.
Try rewriting: 
Multiplying series together: (just for finding first few terms):
Find the first 4 terms of: 
When integrating something that you can't do analytically, convert to series:
Approximate 
by using 4
terms in a power series.
Binomial: 
,
except this only works
for 
being a positive integer. The
REAL formula is:
Find the power series for 
and
.