Calculus Section 9.9 Geometric Power Series -Find a geometric power series that represents a function. -Construct a power series using series operations. Homework: page 674 #’s 5, 8, 9, 10, 11, 12 The final type of series expansion is based off of the geometric series. Review of geometric series: write a geometric series for a1 = 2 and r = 3. Write a geometric series for a1 = 2 and r = -3. Write a geometric series for a1 = 1 and r = x. For a geometric series to converge, the radius for the series must be _________________. If we borrow the a summation formula for a convergent geometric series, sum 1 , we can write a power series for some 1 r types of functions. Examples) Find a Power Series and Interval of Convergence for Each Function 1 1 f ( x) f ( x) c=0 c=2 1 2x 4 x f ( x) 3 ,c=1 3 2x Random Examples of Power/Taylor/Maclaurin Series) x h(x) h’(x) h’’(x) h’’’(x) 1 10 5 6 -2 2 6 4 -7 8 3 -3 -2 9/4 3 Write a second-degree Taylor Polynomial for f about x = 2, and use it to approximate f(2.3). If the third derivative of f satisfies the inequality f ''' ( x) 9 for all x in the interval, use the Lagrange error bound to find an interval [a,b] such that a ≤ f(2.3) ≤ b. If the first 4 terms of the Taylor expansion for f(x) about x = 0 are 15 4 x 3 2 x 2 x 3 , then what is f '''(0) ? 2