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Section 9.8 and 9.9: Power Series

Section 9.8 and 9.9: Power Series. A Polynomial Series. Consider the Polynomial Series:. Investigate the partial sums of the sequence and compare the results to on a graph. Window: and . A Polynomial Series. Consider the Polynomial Series:.

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Section 9.8 and 9.9: Power Series

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  1. Section 9.8 and 9.9: Power Series

  2. A Polynomial Series Consider the Polynomial Series: Investigate the partial sums of the sequence and compare the results to on a graph. Window: and

  3. A Polynomial Series Consider the Polynomial Series: Investigate the partial sums of the sequence and compare the results to on a graph. Window: and

  4. A Polynomial Series Consider the Polynomial Series: Investigate the partial sums of the sequence and compare the results to on a graph. Window: and

  5. A Polynomial Series Consider the Polynomial Series: Investigate the partial sums of the sequence and compare the results to on a graph. Window: and

  6. A Polynomial Series Consider the Polynomial Series: Investigate the partial sums of the sequence and compare the results to on a graph. Window: and

  7. A Polynomial Series Consider the Polynomial Series: Investigate the partial sums of the sequence and compare the results to on a graph. Window: and

  8. A Polynomial Series Consider the Polynomial Series: On , , , , , ,… converges to . The polynomial series is a good approximation of on . Window: and The sequence of polynomials converges to a rational expression. Why?

  9. A Polynomial Series Consider the Polynomial Series: This is the only interval the equation is true. (Same as the graphs.) The series is a geometric series. The first term is 1. The constant ratio is . Therefore the sum is when . So: when .

  10. Power Series Centered at x=0 An expression of the form: is a power series centered at . The interval of convergence is the domain (values for ) for which the series converges. (A series of powers of x)

  11. Example 1 Find a power series to represent and give its interval of convergence. The initial term is . The expression is the sum of a Geometric Series: The constant ratio is . Generate the Series: So the interval of convergence is: Since the series is Geometric, the series converges when :

  12. Example 2 Find a power series to represent and give its interval of convergence. The same as the last series except it has . The expression is the sum of a Geometric Series: Generate the Series with the previous series by replacing the ’s with : So the interval of convergence is: Since the series is Geometric, the series converges when :

  13. Example 3 Find a power series to represent and give its interval of convergence. The initial term is . The expression is the sum of a Geometric Series: The constant ratio is . Generate the Series: So the interval of convergence is no longer centered at 0: Since the series is Geometric, the series converges when :

  14. Power Series Centered at x=a An expression of the form: is a power series centered at . The interval of convergence is the domain (values for ) for which the series converges. (A series of powers of x-a)

  15. Theorem 1: Term-by-Term Differentiation If converges for , then the series obtained by differentiating the series for term by term, converges for and represents on that interval. If the series for converges for all , then so does the series for . Not so Straightforward.

  16. Theorem 1: Term-by-Term Differentiation That theorem can be confusing. What it says is that if… • A power series can be differentiated term by term to form a new series. • The new series will converge to the derivative of the function represented by the original series. • The new series will at least converge on the same interval as the original series. This gives a way to generate new connections between functions and series. Can it converge at a larger interval? Does it include the endpoints? We will not prove this. It is assumed to be true.

  17. Example Find a power series to represent . Notice: for We already know: To find the power series, we differentiate both sides of the equation piece by piece: The last theorem guarantees this series will AT LEAST converge on the same interval as : The series is no longer Geometric. It could converge on a larger interval.

  18. Example: Check the Endpoints Find a power series to represent . For at least We know: Let’s check the endpoint of the interval of convergence to see if the resulting series converges. If : Graph the partial sums: Each Successive term in the sequence of partial sums is outside the two previous terms in this sequence . The Series Diverges

  19. Example: Check the Endpoints Find a power series to represent . For at least We know: The endpoint is not in the interval of convergence. Now check the other endpoint . If : Notice: By the nth Term Test, the Series Diverges.

  20. Example: Conclusion Find a power series to represent . A Power Series to represent that rational expression is: The interval of convergence is The following equation is true for :

  21. Theorem 2: Term-by-Term Integration If converges for , then the series obtained by integrating the series for term by term, converges for and represents on that interval. If the series for converges for all , then so does the series for the integral. Not so Straightforward.

  22. Theorem 1: Term-by-Term Integration That theorem can also be confusing. What it says is that if… • A power series can be integrated term by term to form a new series. • The new series will converge to the integral of the function represented by the original series. • The new series will at least converge on the same interval as the original series. This gives a way to generate new connections between functions and series. Can it converge at a larger interval? Does it include the endpoints? We will not prove this. It is assumed to be true.

  23. Example 1 Find a power series to represent . Notice: for We already know: To find the power series, we integrate both sides of the equation piece by piece: Solve for C.

  24. Example 1: Solve for C What happens to the after you integrate both side of the power series? Let . The is equal to 0. Now go back to the problem.

  25. Example 1: Check the Endpoints The series below converges for at least : The series is no longer Geometric, the interval of convergence could have changed. Let’s check the endpoint of the interval of convergence to see if the resulting series converges. Therefore, does not result in a convergent series and is not in the interval of convergence. This series is the opposite of the Divergent Harmonic Series.

  26. Example 1: Check the Endpoints The series below converges for at least : The endpoint is not in the interval of convergence. Now check the other endpoint . This is the Alternating Harmonic Series. We know the Alternate Harmonic Series converges. So the series above also converges for . Thus, is in the interval of convergence.

  27. Example 1: Conclusion Find a power series to represent . A Power Series to represent that natural log expression is: The interval of convergence is The following equation is true for : And since the equation above holds for , we now know the value for the Alternating Harmonic Series:

  28. The Alternating Harmonic Series The Alternating Harmonic Series converges to :

  29. Example 2 Find a power series to represent . Notice: We already know: for To find the power series, we integrate both sides of the equation piece by piece: Solve for C.

  30. Example 2: Solve for C What happens to the after you integrate both side of the power series? Let . The is equal to 0. Now go back to the problem.

  31. Example 2 Find a power series to represent . Notice: We already know: for To find the power series, we integrate both sides of the equation piece by piece: The last theorem guarantees this series will AT LEAST converge on the same interval as :

  32. Example 2: Check the Endpoints The series below converges for at least : The series is no longer Geometric, the interval of convergence could have changed. Let’s check the endpoint of the interval of convergence to see if the resulting series converges. Each Successive term in the sequence of partial sums is inside the two previous terms in this sequence. Graph the partial sums: The Series Converges and is in the interval of convergence

  33. Example 2: Check the Endpoints The series below converges for at least : The endpoint IS in the interval of convergence. Now check the other endpoint . This is the opposite of the alternating sequence from the other endpoint. Thus, the series converges and is in the interval of convergence.

  34. Example: Conclusion Find a power series to represent . A Power Series to represent that inverse tangent expression is: The interval of convergence is The following equation is true for : And since the equation above holds for , we now know the following:

  35. Approximating Pi with a Power Series From the previous example we know: Notice what happens when we multiply by 4. With a Power Series, we have found a way to approximate Pi. Test the series by finding partial sums to confirm the result.

  36. Closure Interval of Convergence is a challenging topic. In later sections we will learn actual tests to determine if a series converges. The biggest thing we should be concerned with is our ability to make new series from another series. And the possibility the series will be better than the original (i.e. it will converge for more values of x).

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