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Question 3. Taylor series (13 marks) (a) Find the radius and interval of convergence of the power series ?_{n=0}^? rac{n^2}{2^n} x^n. [3 marks] (b) Find the Taylor series (including a formula for the general term) of the following functions at x = 0 and determine their interval of convergence. i. f(x) = rac{1}{3-x} ii. f(x) = rac{1}{(1-x)^3} iii. f(x) = ln(3 - x) (Hint. Take the derivative) [6 marks] (c) Let c be the last non-zero digit of your Monash student ID number and consider the function f(x) = rac{1}{x^2+cx}. Use Mathematica to calculate the Taylor polynomial of degree 5 for f(x) at x = 1. Use Mathematica to plot f(x) for 0 ? x ? 2, as well as the Taylor polynomials of degrees 1, 2 and 3 for f(x) at x = 1. [2 marks] Remark. Approximately one-ninth of you should be pleasantly surprised by your Taylor series! (d) In the lectures, we deduced that the Taylor series for tan^{-1}(x) at x = 0 is given by x - rac{x^3}{3} + rac{x^5}{5} - rac{x^7}{7} + ? + (-1)^{n+1} rac{x^{2n-1}}{2n-1} + ?. Combining this equation with the fact that ? = 4 tan^{-1}(1), we obtain a series for ?. Use Mathematica to calculate the 1000th partial sum of the series to ten decimal places. How many of those ten decimal places agree with the decimal expansion of ?? [2 marks]

          Question 3. Taylor series (13 marks)

(a) Find the radius and interval of convergence of the power series ?_{n=0}^? rac{n^2}{2^n} x^n. [3 marks]

(b) Find the Taylor series (including a formula for the general term) of the following functions at x = 0 and determine their interval of convergence.

i. f(x) = rac{1}{3-x}

ii. f(x) = rac{1}{(1-x)^3}
iii. f(x) = ln(3 - x) (Hint. Take the derivative)

[6 marks]

(c) Let c be the last non-zero digit of your Monash student ID number and consider the function f(x) = rac{1}{x^2+cx}. Use Mathematica to calculate the Taylor polynomial of degree 5 for f(x) at x = 1. Use Mathematica to plot f(x) for 0 ? x ? 2, as well as the Taylor polynomials of degrees 1, 2 and 3 for f(x) at x = 1. [2 marks]

Remark. Approximately one-ninth of you should be pleasantly surprised by your Taylor series!

(d) In the lectures, we deduced that the Taylor series for tan^{-1}(x) at x = 0 is given by

x - rac{x^3}{3} + rac{x^5}{5} - rac{x^7}{7} + ? + (-1)^{n+1} rac{x^{2n-1}}{2n-1} + ?.

Combining this equation with the fact that ? = 4 tan^{-1}(1), we obtain a series for ?. Use Mathematica to calculate the 1000th partial sum of the series to ten decimal places. How many of those ten decimal places agree with the decimal expansion of ?? [2 marks]

Question 3. Taylor series (13 marks)

(a) Find the radius and interval of convergence of the power series ?n=0^? racn^22^n x^n. [3 marks]

(b) Find the Taylor series (including a formula for the general term) of the following functions at x = 0 and determine their interval of convergence.

i. f(x) = rac13-x

ii. f(x) = rac1(1-x)^3
iii. f(x) = ln(3 - x) (Hint. Take the derivative)

[6 marks]

(c) Let c be the last non-zero digit of your Monash student ID number and consider the function f(x) = rac1x^2+cx. Use Mathematica to calculate the Taylor polynomial of degree 5 for f(x) at x = 1. Use Mathematica to plot f(x) for 0 ? x ? 2, as well as the Taylor polynomials of degrees 1, 2 and 3 for f(x) at x = 1. [2 marks]

Remark. Approximately one-ninth of you should be pleasantly surprised by your Taylor series!

(d) In the lectures, we deduced that the Taylor series for tan^-1(x) at x = 0 is given by

x - racx^33 + racx^55 - racx^77 + ? + (-1)^n+1 racx^2n-12n-1 + ?.

Combining this equation with the fact that ? = 4 tan^-1(1), we obtain a series for ?. Use Mathematica to calculate the 1000th partial sum of the series to ten decimal places. How many of those ten decimal places agree with the decimal expansion of ?? [2 marks]

Added by Jeff R.

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