1. Do Now: #18 and 20 on p.466 Find the interval of convergence and the function of x represented by the given geometric series. This series will only converge when : Interval of convergence:

2. Do Now: #18 and 20 on p.466 Find the interval of convergence and the function of x represented by the given geometric series. Sum of the series: So, this series represents the function Graphical support?

3. Do Now: #18 and 20 on p.466 Find the interval of convergence and the function of x represented by the given geometric series. This series will only converge when : Interval of convergence:

4. Do Now: #18 and 20 on p.466 Find the interval of convergence and the function of x represented by the given geometric series. Sum of the series: So, this series represents the function Graphical support?

5. Section 9.1b

6. Similar Problems: #42 and 44 on p.468 Find a power series to represent the given function and identify its interval of convergence. Compare to: Series: The series converges when : Interval of convergence:

7. Similar Problems: #42 and 44 on p.468 Find a power series to represent the given function and identify its interval of convergence. Compare to: Series: The series converges when : Interval of convergence:

8. Finding a Power Series by Differentiation Given that 1/(1 – x) is represented by the power series find a power series to represent Notice that this is the derivative of this !!! With the same interval of convergence: Graphical support?

9. Theorem: Term-by-Term Differentiation If converges for, then the series obtained by differentiating the series for term by term, converges on the same interval and represents on that interval.

10. Finding a Power Series by Integration Given that find the power series to represent With the same interval of convergence:

11. Theorem: Term-by-Term Integration If converges for, then the series obtained by integrating the series for term by term, converges on the same interval and represents on that interval.

12. Exploration 3 on p.465 1. 2. Since this function is its own derivative and takes on the value 1 at x = 0…. perhaps it is the exponential function? 3. 4.

13. Exploration 3 on p.465 5. 6. 7. It appears that the first three partial sums may converge on (–1, 1)… The actual interval of convergence is all real numbers!!!