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Calculus II (MAT 146) Dr. Day Wednes day November 20, 2013

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Power Series ConvergencePower Series ConvergencePower Series Convergence

Calculus II (MAT 146)Dr. Day Wednesday November 20, 2013

- Transforming Series into Functions
- Power Series
- Coefficients
- Applying the Ratio Test to Power Series
- Radius of Convergence
- Interval of Convergence
- Assignments and Announcements

MAT 146

Power Series

- x is a variable.
- The cn’s are constants, called the coefficients.
- For any fixed value of x, we can test the series for convergence.

MAT 146

Power Series

- The sum of the series is a function with domain the set of all x values for which the series converges.
- The function seems to be a polynomial, except it has an infinite number of terms.

MAT 146

Power Series: Example

- If we let cn = 1 for all n, we get a familiar series:

- This geometric series has common ratio x and we know the series converges for |x| < 1.
- We also know the sum of this series:

MAT 146

Generalized Power Series

- This is called:
- a power series in (x – a), or
- a power series centered at a, or
- a power series about a.

MAT 146

Power Series Convergence

- For what values of x does this series converge?
- Determine its Radius of Convergence and its Interval of Convergence.

MAT 146

Power Series Convergence

- For what values of x does this series converge?
- Determine its Radius of Convergence and its Interval of Convergence.

MAT 146

Power Series Convergence

- For what values of x does this series converge?
- Use the Ratio Test to determine values of x that result in a convergent series.

MAT 146

Power Series Convergence

- For what values of x does this series converge?
- Use the Ratio Test to determine values of x that result in a convergent series.

MAT 146

Power Series Convergence

- For what values of x does this series converge?
- Determine its Radius of Convergence and its Interval of Convergence.

MAT 146

- For what values of x does this series converge?
- Determine its Radius of Convergence and its Interval of Convergence.

MAT 146

- For what values of x does this series converge?
- Determine its Radius of Convergence and its Interval of Convergence.

MAT 146

- For what values of x does this series converge?
- Determine its Radius of Convergence and its Interval of Convergence.

MAT 146

Geometric Power Series

- If we let cn = 1 for all n, we get a familiar series:

- This geometric series has common ratio x and we know the series converges for |x| < 1.
- We also know the sum of this series:

MAT 146

Geometric Power Series

MAT 146

Geometric Power Series

MAT 146

Geometric Power Series

MAT 146

Assignments

Homework Tasks

- Today: Quiz #8 returned, solutions online
- This Week: 11.8 (Today)
- After Break: 11.9 (Wed 12/4) and 11.10 (Fri 12/6)
- By the Day of Semester Exam: Four Review Assignments

Also

- Test #4: Tomorrow and Fri (11/21 and 11/22)
- Help Session Tonight: STV 214, 6:15 pm – 7:30 pm
- Check Your Semester Exam Schedule!
- Semester Extra-Credit Option: Due December 2

MAT 146

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