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Name__________________________________ Period ________ 1) Geometric Series: x n Calculus Lesson 9.1 Day 2 1 x x 2 ... x n ... n 0 Sigma Series 1 1 x Sum Formula Find the value(s) of x for which the given geometric series converges. Find the sum of the series. Power Series are series of algebraic functions. We use power series to approximate transcendental functions. Power Series An expression of the form c x n 0 n n c0 c1 x c2 x 2 ... cn x n ... is a power series centered at x 0 . An expression of the form c x a n 0 n n c0 c1 x a c2 x a ... cn x a ... is a power series 2 centered at x a . The term cn x a is the nth term; the number a is the center. n 2.) Find a power series that represents 1 on (-1,1). 1 x n 3.) Find a power series that represents 1 on (-1/2,1/2). 1 2x 4.) Find a power series that represents x on (-1,1). 1 x 5.) Find a power series that represents 1 and give its interval of convergence. 5 x 6.) Find a power series that represents 1 and give its interval of convergence. 3x Term-By-Term Differentiation 1 1 x x 2 x 3 x 4 ... x n ... on the interval (-1,1), find a power series to represent 7.) Given that 1 x 1 . 2 1 x Term-By-Term Integration. 1 n 1 x x 2 x3 x 4 x5 ... x ... on the interval (-1,1), find a power series to 8.) Given that 1 x represent ln(1 x) . We have been looking at graphs of power series. We can also use power series to evaluate expressions. ln(2) 0.6931471806 Let x 1 and complete the table. Number of Terms 1 Expression Approximation x Numerical Approximation 1 ln(2) approximation 2 x - x2/2 0.5 0.1931471806 3 x - x2/2 + x3/3 0.8333 0.1401861527 4 x - x2/2 + x3/3 - x4/4 0.58333 0.1098138472 5 6 0.3068528194 Additional Questions: Find the interval of convergence and the function of x represented by the geometric series. 9.) 3 x n n 0 x 1 10.) 5 n 0 2 n 64.) Use differentiation to find a series for f ( x) n 11.) n n 0 2 1 x cos 3 . (Use the work in #7 to help.) Challenge Question: x 2 x3 x 4 xn If f x 1 x ... ... , what well-known function do you suppose f to be? 2! 3! 4! n! x