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7) Find the Taylor series generated by fat x = 1 f(x) = 2x³ + x² + 3x − 8

7) Find the Taylor series generated by fat x = 1 f(x) = 2x³ + x² + 3x − 8

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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7) Find the Taylor series generated by fat x = 1 f(x) = 2x³ + x² + 3x - 8 8) Find the Taylor series generated by fat x = 1 f(x) = 1/x² 9) Find the Taylor series generated by fat x = 0 cos²x (Hint: cos²x = (1 + cos2x)/2.)
Transcribed Image Text:7) Find the Taylor series generated by fat x = 1 f(x) = 2x³ + x² + 3x - 8 8) Find the Taylor series generated by fat x = 1 f(x) = 1/x² 9) Find the Taylor series generated by fat x = 0 cos²x (Hint: cos²x = (1 + cos2x)/2.)
Example: Find the values of x for which the power series converges. Apply the Ratio Test Σ(-1)-1.x" Un+1 Un n=1 = n=1 Un+1 Un n xn+1 n n+ 1 x x2n-1 Σ(1)"-1, 2n 1 = At x = 1, we get the alternating harmonic series which converges. At x = -1, we get the negative of the harmonic series; it diverges. 0 -1 < x≤ 1 Example: Find the values of x for which the power series converges. Apply the Ratio Test = X . = At x = 1 the series x2n+1 2n - 1¹ 2n + 1x2n-1 At x = -1 the series 2 3 n n+1 -| x | → |x| The series converges absolutely for x < 1. It diverges if |x| > 1 = 2n - 1 218² 2n + 1 →x² The series converges absolutely for x² < 1. It diverges for x² > 1 0 -1 ≤ x ≤ 1 x
Transcribed Image Text:Example: Find the values of x for which the power series converges. Apply the Ratio Test Σ(-1)-1.x" Un+1 Un n=1 = n=1 Un+1 Un n xn+1 n n+ 1 x x2n-1 Σ(1)"-1, 2n 1 = At x = 1, we get the alternating harmonic series which converges. At x = -1, we get the negative of the harmonic series; it diverges. 0 -1 < x≤ 1 Example: Find the values of x for which the power series converges. Apply the Ratio Test = X . = At x = 1 the series x2n+1 2n - 1¹ 2n + 1x2n-1 At x = -1 the series 2 3 n n+1 -| x | → |x| The series converges absolutely for x < 1. It diverges if |x| > 1 = 2n - 1 218² 2n + 1 →x² The series converges absolutely for x² < 1. It diverges for x² > 1 0 -1 ≤ x ≤ 1 x
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