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LBS 119 – REVIEW for Exam # 2.
Power Series
•
Be prepared to find the CENTER, RADIUS and INTERVAL of convergence for a power series.
•
If you have a GEOMETRIC power series, then this is just a case of finding where the absolute
value of r is less than one.
•
For NONGEOMETRIC power series, we can start by using the RATIO or ROOT test (usually
ratio test) to find the RADIUS of convergence… but remember, if asked to find the INTERVAL
of convergence, we must check the ENDPOINTS of the interval separately (using OLD
convergence / divergence tests from Exam 1).
Be prepared to use them if needed…
Remember:
PLAN DR RIGHT.
•
REMINDER:
The Ratio and Root Tests are both tests for ABSOLUTE convergence.
(Always
use absolute value.)
RATIO TEST:
Find
1
lim
n
n
n
a
a
+
ROOT TEST:
Find
lim
n
n
n
a
a
In both
tests,
If the limit <1, the series CONVERGES;
If the limit >1, the series DIVERGES;
If the limit = 1, the test is inconclusive.
(You must use a different test.)
•
Remember, the DERIVATIVE and INTEGRAL of a power series have the same RADIUS of
convergence as the original series!
(But convergence might differ at the ENDPOINTS.)
Representation of a Function by a Power Series.
•
We know that Geometric Series have the following form:
(YOU NEED TO KNOW THIS)
0
1
1
n
n
a
ar
for r
r
a
=
= <

.
If we want to find a power series for a function, we can sometimes
rewrite it in the form
1
a
r

.
Then, when we know the values of
a
and
r
, we can plug them into
the formula above and come up with a power series for the function.
Remember, if you need a
different center
x = c, then you need r to have an
x
c

in it.
•
Sometimes partial fractions are needed to separate factors in the denominator first.
•
You could also use long division to find a power series for any rational function.
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 Spring '08
 HanniNichols

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