Power Tools for Power Series

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Power Tools for Power Series
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The power series labs
use some terms in a specific way. Its
worthwhile to see if everyone is up on (or down
with)this terminology.
xn
an
n!
2n1
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Terms to know
Sequence a function defined on the set of
positive integers f(n) an Series a
function formed from a sequence the values of
the sequence serve as the coefficients of
subsequent terms.
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Limits again?
A sequence has a limit if it satisfies the
following formal criterion
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Terms to know
A sequence that has a limit is convergent. A
sequence that has no limit is divergent.
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Limit Theorem
If a function and a sequence have f(n) an for
every positive integer n, then the sequence and
the function have the same limit.
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Squeeze Theorem
Let an, bn and cn be sequences. If an and bn
have the same limit L as n and there exists an
integer N such that an cn bn for all ngtN,
then the limit of the sequence cn is likewise L.
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Properties of Sequences
A monotonic sequence has terms that are either
entirely non-decreasing or non-increasing.
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Bounding Properties
A sequence is bounded above if there is a real
number M such that an M for all n. A sequence
is bounded below if there is a real number N such
that N a n for all n.
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More than one bound?
The least upper bound is the bound that is
smaller than all the others.
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Properties of Sequences
If a sequence is bounded above and below, then it
is bounded. A sequence that is bounded and
monotonic converges and therefore has a limit.
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An Infinite Series
Like the Energizer Bunny, it just keeps going!
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Partial Sums
If the sequence Sn converges to the sum S, then
the series S an converges and the limit is S.
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Power Series

• For the variable x, the constant c and the series
  an, the power series centered at c is defined
  as

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Approximating a function
For example, you have already worked with the
power series approximation for f(x) sin x,
given by the Taylor series
Hmm, odd powers only and the sin function has
odd symmetry hmm .
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Convergent or Divergent?
f(x) sin x
First term x
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Convergent or Divergent?
f(x) sin x
Add the 2nd term - x3/3
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Convergent or Divergent?
f(x) sin x
Add more terms
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Convergent or Divergent?
f(x) sin x
Pretty soon, the approximation is very good!
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Convergent or Divergent?
f(x) sin x
The higher degree the polynomial, the farther we
can go from the origin!
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Illustrating

• A power series can have terms that alternate in
  sign and the series can still converge.
• A convergent series need not be monotonic.

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But some functions are more difficult

• The power series expansion for f(x) 1/(1-x)
  is interesting because it lets you evaluate f(x)
  without worrying about the discontinuity (which
  the physicists call a singularity)

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f(x) 1/(1-x)
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f(x) 1/(1-x)
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f(x) 1/(1-x)
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This hasnt gone so well
Not bad on (1, 1), but the approximation
doesnt cut it for xlt-1 or xgt1.
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Suggesting that some series only converge over
limited intervals

• The radius of convergence of a series centered at
  the constant c is that value of R gt 0 such that
  the series converges for x - c lt R and
  diverges for x c gt R.

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So we need some tests for convergence
Radius of convergence 1