WebNov 21, 2016 · The series of course depends on where you expand around. Taking x = 0 as an example: f ( x) = 1 ( x + 3) 2 = ∑ n = 0 ∞ f ( n) ( 0) x n n!, where f ( n) ( x) = ( − 1) n ( n + 1)! ( x + 3) n + 2. So the series is 1 ( x + 3) 2 = ∑ n = 0 ∞ ( − 1) n ( n + 1)! ( 3) n + 2 x n n! = ∑ n = 0 ∞ ( − 1) n ( n + 1) ( 3) n + 2 x n, WebEach term is representable by a power series by using the geometric series theorem. Notice that 1 1 − 2 x = ∑ n = 0 ∞ 2 n x n, x < 1 / 2, and that 1 1 + x = ∑ n = 0 ∞ ( − 1) n x n, x < 1. Combine appropriately to get your desired power series expansion. Share Cite answered Mar 29, 2013 at 20:47 ncmathsadist 48.3k 3 78 128 Add a comment 1
Solved Find a power series representation for the function ... - Chegg
WebFeb 6, 2024 · Calculus Power Series Introduction to Power Series 1 Answer Andrea S. Feb 6, 2024 ln(1 + x) = ∞ ∑ n=0( −1)n xn+1 n +1 with radius of convergence R = 1 Explanation: Start from: ln(1 + x) = ∫ x 0 dt 1 +t Now the integrand function is the sum of a geometric series of ratio −t: 1 1 + t = ∞ ∑ n=0( − 1)ntn so: ln(1 + x) = ∫ x 0 ∞ ∑ n=0( −1)ntn WebFinding function from power series by integrating AP.CALC: LIM‑8 (EU) , LIM‑8.D (LO) , LIM‑8.D.6 (EK) , LIM‑8.G (LO) , LIM‑8.G.1 (EK) Google Classroom About Transcript When a power series S₁ is an antiderivative of a geometric series S₂, we can find the function represented by S₁ by integrating the expression for S₂. Sort by: Top Voted Questions scratchy throat when breathing
Solved (a) Use differentiation to find a power series Chegg.com
WebJun 22, 2024 · This is fairly similar to 1 1 − x, for which we know a power series: 1 1 − x = 1 + x + x2 +... = ∞ ∑ k=0xk The radius of convergence for this power series is x ∈ ( − 1,1). While it would be easy to say that 1 (1 − x)2 = ( ∞ ∑ k=0xk)2 This is not a valid representation of a power series. Usually, some power series arise from derivatives. WebExample 2 : Find a power series representation of the function 1 7 + 2 x, and determine for which x it would be defined. Solution 1: Replace x (in our original f ( x) before the … WebScience Advanced Physics (a) Use differentiation to find a power series representation for f (x) = 10 7= 0 f (x) = m What is the radius of convergence, R? (b) Use part (a) to find a … scratchy timbre