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PART II10.2 Power Series
Do not use a calculator! Do your work for Part II on a separate sheet of paper. Give a
concluding statement in mathematical sentence or a complete (nonmathematical) written
sentence as appropriate for each part. Staple your work for Part I and Part II together
before you turn it in.
1.
Goal
for this problem
: Use power series to approximate
with an error less
than 0.0001.
Begin with a power series representation for a function that we do know:
a.)
Use this geometric power series to find a power series for
using substitution.
Write this series in closed form, i.e. expressed in series notation with the nth term.
What
is its interval of convergence? Why?
•
*
(Thinking ahead:
).
•
We want to express
as a sum of a power series and find the interval of
convergence. To do this using substitution then we need to replace the
x
in
by
:
*
•
Now we need to determine the interval of convergence for the power series
*
Observe that
is a geometric series where
. Thus the series
converges absolutely when
.
arctan(.5)
1
1

x
=
x
n
=
1
+
x
+
x
2
+
x
3
+
...
+
n
=
0
∞
∑
x
n
+
...
for
x
<
1
1
1
+
x
2
1
1
+
x
2
dx
=
arctan
x
+
C
∫
1
1
+
x
2
1
1

x

x
2
1
1
+
x
2
=
1
1
+ 
x
2
( )
=
1
+ 
x
2
( )
+ 
x
2
( )
2
+ 
x
2
( )
3
+
...
=
1

x
2
+
x
4

x
6
+
...
=

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 Spring '08
 Any
 Calculus, Power Series

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