1. Find the Taylor series expansion of the following and determine the circle of convergence. (a) about zo i (b) 22 about 20 = 1 (c) ze about 20 = 1 (d) (-2)3 about 20 = 0 (Hint: Work with 2 first and take derivatives.)
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- How would I find if the sequences an=nπ cos(nπ) and an= (n2)(1/n) are convergent/divergent?5. Suppose that the power series E Cnx" converges when x = 3 and diverges when x = 6. Which of the following n=0 statements is certain to be true? (a) None of these statements is certain to be true. (b) Σ Cn(-4)" is convergent. n=0 (c) Σ Cn(-3)" is convergent. n=0 (d) > Cn(-2)" is convergent. n=0 (e) 2 Cn(-6)" is divergent. n=0(4.x – 12)" - 3. Consider the power series n=1 (-3)2+"(n² + 1)* (a) Determine the radius of convergence. (b) Determine the interval of convergence.
- Need help with bottom 3 partsExercise 4.1 Find the radius and interval of convergence for the following series. 1. En-1n+1 2. En-1)"x" 3. En=1n n=1 4. En=12(x- 3)" 5. ΣΗ- 6. Σ-1 (-1)"x+1 (n+1)! 7. En=1 (-1)"(x-2)" 8. Ln-1 2 (n+1) 9. En=1 10. E-1n(x+ 5)"12. Find the radius and interval of convergence of the power series n=0 (x-4) √5" n 18. Find the 3th -degree Taylor polynomial centered at x=8 for the function f(x)=√x. Use the polynomial to approximate 3/8.02. Compare the value to the value obtained by your calculator. 19. Find the first three non-zero terms of the Taylor series for f(x) = secx centered at a=. Please show all work. 20. How large must n be to approximate values of sin x within 0.01 on the interval (-1,1) using the series sin(x) = 8 Σ (−1)"x2n+1 ? (2n+1)! n=0
- 6. Consider a power series of the form Σen (x-a)n n=0 with radius of convergence R. If a 0 and a #r, which of the following are possible intervals of convergence? (a) [-R, R) (b) (R-a, R+ a) (c) (a R/2, a + R/2) (d) (a R, a + R] (e) None of the above.Q6. (1) Given log (1+t) = t-t² +t²-t ² + --- 2 3 4 valid for It3. Consider the function f defined by f(x) = b, 0Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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