9.4 Power Series II: Geometric Series
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1 9.4 Power Series II: Geometric Series A particularly important skill to develop for the AP eam, other than checking that you re in RADIAN mode, is to represent certain types of rational functions as a geometric series. Rather than producing the power series using Taylor s Rule, you will want to develop the series by manipulating a geometric series or, in some cases, using Long Division. Eample : First we ll do a quick review of geometric series. Geometric series are formed by multiplying by a common ratio r. (a) Suppose I told you to start with a and to let would the sum be? r, what geometric series would you write? What 3 (b) What if a and r? 3 (c) What if a and r? Eample : Verify your answer from Eample (c) by finding the power series for (a) using Taylor s Rule (b) using LOOOONG DIVISION. (c) Find the radius and interval of convergence. Verify by graphing. centered at c 0 by Page of 6
2 Eample 3: Find a power series for centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Eample 4: Find a power series that represents centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Eample 5: Find a power series for f( ) centered at c 0, then find the interval of convergence. Find the first four nonzero terms and the general term. Page of 6
3 When you replace with a multiple of, beware a change in the radius and interval of convergence... Eample 6: Find a power series that represents centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Eample 7: Find a power series for g ( ) 4 centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Sometimes we cannot center our function at 0. In this case, we must try to rewrite our function with the new center showing. Eample 8: Find a power series that represents centered at c, then find the interval of convergence. Include the first four nonzero terms and the general term. Page 3 of 6
4 Eample 9: 5 (A bit of a booger) Find a power series for h ( ), centered at c, then find the interval of convergence. Include the first four nonzero terms and the general term. We can integrate or differentiate a power series to obtain a new series. When we do this, the radius of convergence will be the same, but the interval may change. In this case, we must retest the endpoints. Eample 0: Find a power series that represents radius of convergence? Interval of convergence? centered at c 0. Hint: what is d d? What is the Eample : Find a power series that represents ln radius of convergence? Interval of convergence? centered at c 0. Hint: what is d? What is the Page 4 of 6
5 Eample : (Similar to 008 BC6B ) Let f be the function given by f. (a) Write the first four nonzero terms and the general term of the Taylor series for f about 0. (b) Does the series found in part (a), when evaluated at, converge to f? Eplain why or why not. (c) The derivative of arctan is arctan about 0.. Write the first four nonzero terms of the Taylor series for (d) Use the series found in part (c) to find a rational number A such that your answer. 3 A arctan. Justify 4 0 Page 5 of 6
6 Besides finding the sum of an infinite, convergent geometric series (and the telescoping series), there is one last way we are epected to find such sums: by recognizing a given infinite, convergent series as a Taylor series evaluated at a particular value of. Eample 3: By recognizing each series as a Taylor series evaluated at a particular value of, find the sum of each of the following infinite, convergent series. 4 6 (a)! 4! 6! 3 4 ln 4 ln 4 ln 4 (b) ln 4! 3! 4! (c) ! 6 5! 6 7! 6 Page 6 of 6
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