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(3) (a) Show that 1 1 1+g² = Σ(-1)"x²n if |x| < 1. You may use the formula 1-2 n=0 Σz" if |2|< 1. n=0 8 (b) Prove that arctan(x) = Σ n=0 arctan(x) + C from calculus. (-1)¹x²n+1 2n + 1 for x < 1. You may use that +/1 (c) Show that the formula in part (b) also holds if |x| you get if you plug in x = 1 in part (b)? 1 1 + x² dx = 1. What formula for 7 do -

(3) (a) Show that 1 1 1+g² = Σ(-1)"x²n if |x| < 1. You may use the formula 1-2 n=0 Σz" if |2|< 1. n=0 8 (b) Prove that arctan(x) = Σ n=0 arctan(x) + C from calculus. (-1)¹x²n+1 2n + 1 for x < 1. You may use that +/1 (c) Show that the formula in part (b) also holds if |x| you get if you plug in x = 1 in part (b)? 1 1 + x² dx = 1. What formula for 7 do -

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(3) (a) Show that ∞ n=0 1 1+x² zª if |z| < 1. - (b) Prove that arctan(x) ∞ n=0 = (−1)″x²n if |x| < 1. You may use the formula -1)”.g2n+1 2n + 1 1 2 1 for x < 1. You may use that J 1+x² dx n=0 arctan(x) + C from calculus. (c) Show that the formula in part (b) also holds if |x| = 1. What formula for do you get if you plug in x = 1 in part (b)?
Transcribed Image Text:(3) (a) Show that ∞ n=0 1 1+x² zª if |z| < 1. - (b) Prove that arctan(x) ∞ n=0 = (−1)″x²n if |x| < 1. You may use the formula -1)”.g2n+1 2n + 1 1 2 1 for x < 1. You may use that J 1+x² dx n=0 arctan(x) + C from calculus. (c) Show that the formula in part (b) also holds if |x| = 1. What formula for do you get if you plug in x = 1 in part (b)?
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