Calculus II

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Calculus II

• Power Series

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Definition
A power series about x 0 is a series of the form
A power series about x a is a series of the form
in which the center a and the coefficients are
constants.
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Example Consider the Geometric Series.
This series converges to on the
interval .
We say that this series is centered at x 0, has
a radius of convergence of 1, and an interval of
convergence of .
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Example Consider the power series
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How to Test a Power Series for Convergence
Step 1 Use the Ratio Test (or nth-root test) to
find the interval where the series converges
absolutely. Ordinarily, this is an open
interval Step 2 If the interval of absolute
convergence is finite, test for convergence or
divergence at each endpoint. Use a Comparison
Test, the Integral Test, or the Alternating
Series Test. Step 3 If the interval of
absolute convergence is the series diverges for
(it does not even
converge conditionally), because the nth term
does not approach zero for those values of x.
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Example Interval of converence.
(a) Find the series radius and interval of
convergence. For what values of x does the
series converge, (b) absolutely and (c)
conditionally.
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Example Interval of converence.
(a) Find the series radius and interval of
convergence. For what values of x does the
series converge, (b) absolutely and (c)
conditionally.
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The Convergence Theorem for Power Series
If
converges for , then it converges
absolutely for all . If the series
diverges for , then it diverges for all
.
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Possible Behavior of
1. There is a positive number R such that the
series diverges for but
converges absolutely for . The
series may or may not converge at either of
the endpoints x a - R and x a R. 2.
The series converges absolutely for every x (R
8). 3. The series converges at x a and
diverges elsewhere (R 0).
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The Term-by-Term Differentiation Theorem
If converges for a - R lt x lt a
R for some R gt 0, it defines a function f
Such a function f has derivatives of all orders
inside the interval of convergence. We can
obtain the derivatives by differentiating the
original series term by term
and so on. Each of these derived series
converges at every interior point of the interval
of convergence of the original series.
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The Term-by-Term Integration Theorem
Suppose that
converges for a - R lt x lt a R (R gt 0). Then
converges for a - R lt x lt a R and
for a - R lt x lt a R.
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The Series Multiplication Theorem for Power Series
If and
converge absolutely for , and
Then converges absolutely to A(x)B(x)
for
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