is a powerCalculus
Lesson
55
Power
Series
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A Taylor Polynomial is a finite form of a Power Series.
The general form of a power series: is a power
series centered at 
.
If 
, the domain of the series is
all 
s for which
converges.
Exactly one of the following is always true for any power series:
1) it converges only at 
.
2) always diverges for all 
s.
3) There exists some 
such
that 
converges
absolutely
and 
diverges.
is the Radius of
Convergence:
c - R c
c + R
Note that the endpoints could be any of (), [], (], [).
We must test to see which one it is.
Examples: Find R for all of the following:
1)
For 
, we can use the ratio test.
and we
need
the result to be 
for
convergence, so 
2) 
For 
, we can use the ratio test.
There are no values for which it converges, so 
3) 
For 
, we can use the ratio test.
Since it converges for all values of 
, infinite radius of
convergence.
There's also the Interval of Convergence:
From (1) above, it's centered about 
with 
, so now we just need
to check 
. Neither 
and 
converge, so the
interval of convergence is 
.
From (2) above, there is no interval of convergence.
From (3) above, the interval of convergence is 
.
4) 
Centered at 
. Use ratio
test to find 
.
Alternating harmonic converges and harmonic does not,
so 
.
5) 
Centered at 
. Use ratio
test to find 
.
Neither 
converges,
so 
.
In the next section, we will be generating many power series.
Each
power series MUST be accompanied by an interval of convergence!!!!
Also, now that we can put things in terms of a power series, we can differentiate and
integrate term by term.
Example: Find the interval convergence of
for
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