# Calculus II (MAT 146) Dr. Day Wednes day November 20, 2013 - PowerPoint PPT Presentation

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Calculus II (MAT 146) Dr. Day Wednes day 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. Use the Ratio Test to Determine Convergence.

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

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### Calculus II (MAT 146)Dr. DayWednesday November 20, 2013

• Transforming Series into Functions

• Power Series

• Coefficients

• Applying the Ratio Test to Power Series

• Interval of Convergence

• Assignments and Announcements

MAT 146

MAT 146

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

### 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?

• 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

MAT 146

MAT 146

MAT 146

### Assignments

• 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