Calculus 9.1

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9.1 Power Series
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What You Will Learn

• All continuous functions can be represented as a
  polynomial
• Polynomials are easy to integrate and
  differentiate
• Calculators use polynomials to calculate trig
  functions, logarithmic functions etc.
• Downfall of polynomial equivalent functions is
  that they have an infinite number of terms.

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Start with a square one unit by one unit
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This is an example of an infinite series.
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This series converges (approaches a limiting
value.)
Many series do not converge
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In an infinite series
a1, a2, are terms of the series.
an is the nth term.
Partial sums
nth partial sum
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Geometric Series
In a geometric series, each term is found by
multiplying the preceding term by the same
number, r.
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Geometric Series
Partial Sum of a Geometric Series Sn a
ar ar2 ar3 arn-1 -r Sn ar
ar2 ar3 arn Sn r Sn a arn
Sn (1 r) a (1 - rn)
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Sum of Converging Series
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Power Series Using Calculator
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Example 1
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Example 2
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The partial sum of a geometric series is
The more terms we use, the better our
approximation (over the interval of convergence.)
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Example of a Power Series
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A power series is in this form
or
The coefficients c0, c1, c2 are constants.
The center a is also a constant.
(The first series would be centered at the origin
if you graphed it. The second series would be
shifted left or right. a is the new center.)
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Once we have a series that we know, we can find a
new series by doing the same thing to the left
and right hand sides of the equation.
This is a geometric series where r-x.
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Example 4
Given
find
We differentiated term by term.
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Example 5
Given
find
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Example 5
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This series would allow us to calculate a
transcendental function to as much accuracy as we
like using only pencil and paper!
p
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Convergent Series
Only two kinds of series converge 1)
Geometric whose r lt 1 2) Telescoping
series Example of a telescoping series the
middle terms cancel out
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Finding a series for tan-1 x

• 1. Find a power series that represents
  on (-1,1)
• Use integration to find a power series that
  represents
• tan-1 x.
• Graph the first four partial sums. Do the graphs
  suggest convergence on the open interval (-1, 1)?
• 4. Do you think that the series for tan-1 x
  converges at x 1?

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Guess the function