Power Series

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Warm Up
Determine the interval of convergence for the
series
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WARM UP

• Determine the sum of the infinite geometric
  series
• Which of the following series converge?
• b)
• c) d)

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Power Series

• and elementary functions

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Consider the series

• Write out the first four terms of the series.
• Does the series converge?
• How do you know?
• What is the sum of the series?
• What if ¼ is replaced by x?

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The function is called an elementary function
and represents the sum of the Power Series
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Other elementary functions that you must know
the Power series for are

• ln(x) (centered at x 1)
• ex (centered at x 0)
• cos(x) (centered at x 0)
• sin(x) (centered at x 0)

You can determine the power series by using the
Taylor polynomial formula until you figure out
the pattern. We have already done ln(x) and ex.
Determine the Power Series for cos(x) and sin(x).
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You can use an elementary function Power Series
to derive other Power Series

• Write the first four non-zero terms and the
  general term for the power series

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You can use an elementary functions Power Series
to derive other Power Series

• Write the first four non-zero terms and the
  general term for the power series

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Power series can be multiplied, divided, added
and subtracted like polynomials.

• Determine the first four nonzero terms and the
  general term of the series

xsin(x)
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• Determine the first four nonzero terms and the
  general term of the series

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• Determine the first four nonzero terms and the
  general term of the series

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You can verify derivatives using Power Series

• Use Power series to show that the derivative of
  sin (x) is cos (x)

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