Edit: After watching the video, it appears the function in question is f (x)kx0, and this is indeed k1 for all x, including x0. Find out how to watch the Power shows in order, as well as where to watch. In order to find a series representation outside this range. Also supporting the statement 001 is a somewhat fundamental definition of exponentiation: xy means start with one, and multiply it by x y times. x 1 x 2 x n 0 ( x 2) n n 0 x ( x 2) n n 0 ( 1) n x 2 n 1 x. Solution 1: Replace x (in our original f ( x) before the video) by x 2, and multiply the expression by x. This series converges when ∣ x ∣ < 1 |x| < 1 ∣ x ∣ < 1. 6.13 Taking the derivative of a power series does not change its radius of convergence. Example 2 : Find a power series representation of the function 1 7 2 x, and determine for which x it would be defined. (Give your power series representation centered at x 0.) provided 1x1 < 1 Viewing Saved Work Revort to Last Response -15 POINTS Find a power series representation for the function find the interval of convergence. F ( x ) = x 2 tan − 1 ( x 3 ) f(x) = x^2 \tan^ \\ģ x 2 ⋅ 1 − ( − x 6 ) 1 = 3 x 2 n = 0 ∑ ∞ ( − x 6 ) n = n = 0 ∑ ∞ ( − 1 ) n 3 x 6 n 2 = 3 x 2 − 3 x 8 3 x 14 − 3 x 20 . Neumann functions and are defined by From the power series representation of Jm (x) we have Thus, using we find Similarly, starting from the power series. Find a power series representation for the function find the interval of convergence. We can obtain power series representation for a wider variety of functions by exploiting the fact that a convergent power series can be differentiated.
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