Essex County College — Division of Mathematics and Physics
1
Lecture Notes #13 — Sakai Web Project Material
We won’t ﬁnish this in class, but I do encourage you to continue working on this on your own.
You should also try using a graphing utility to graph very high degree polynomial .
.. you might
learn something!
Consider the functions
y
=
e

x
2
and
y
=
1
1 +
x
2
.
1. Write the Taylor expansions for the two functions about
x
= 0. What is similar about
these two series? What is diﬀerent?
2. Looking at the series, which function do you predict will be greater over the interval
(

1
,
1)? Graph both and see.
3. Are these functions
even
or
odd
? How might you see this by looking at the series expansion?
4. By looking at the coeﬃcients, explain why it is reasonable that the series for
y
=
e

x
2
converges for all values of
x
, but the series for
y
=
1
1 +
x
2
converges only on (

1
,
1).
1
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 Spring '10
 Ban
 Calculus, Division, Derivative, Taylor Series, Analytic function

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