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Question 3. Taylor series (15 marks) (a) Find the radius and interval of convergence of the power series sum_{n=0}^{infty} frac{n^2}{2^n} x^n. [3 marks] (b) Find the Taylor series (including a formula for the general term) of the function f(x) = frac{1}{1+x} at x = 0 and determine its interval of convergence. [4 marks] (c) Let c be the last non-zero digit of your Monash student ID number and consider the function f(x) = frac{1}{x^2 + c x}. Use Mathematica to calculate the Taylor polynomial of degree 5 for f(x) at x = 1. Use Mathematica to plot f(x) for 0 le x le 2, as well as the Taylor polynomials of degrees 1, 2 and 3 for f(x) at x = 1. [4 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 - x^3/3 + x^5/5 - x^7/7 + cdots + (-1)^{n+1} frac{x^{2n-1}}{2n-1} + cdots Combining this equation with the fact that pi = 4 an^{-1}(1), we obtain a series for pi. 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 pi? [2 marks]

          Question 3. Taylor series (15 marks)

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

(b) Find the Taylor series (including a formula for the general term) of the function f(x) = frac{1}{1+x} at x = 0 and determine its interval of convergence. [4 marks]

(c) Let c be the last non-zero digit of your Monash student ID number and consider the function f(x) = frac{1}{x^2 + c x}. Use Mathematica to calculate the Taylor polynomial of degree 5 for f(x) at x = 1. Use Mathematica to plot f(x) for 0 le x le 2, as well as the Taylor polynomials of degrees 1, 2 and 3 for f(x) at x = 1. [4 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 - x^3/3 + x^5/5 - x^7/7 + cdots + (-1)^{n+1} frac{x^{2n-1}}{2n-1} + cdots

Combining this equation with the fact that pi = 4	an^{-1}(1), we obtain a series for pi. 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 pi? [2 marks]

$Question 3. Taylor series (15 marks) (a) Find the radius and interval of convergence of the power series sumn=0^infty fracn^22^n x^n. [3 marks] (b) Find the Taylor series (including a formula for the general term) of the function f(x) = frac11+x at x = 0 and determine its interval of convergence. [4 marks] (c) Let c be the last non-zero digit of your Monash student ID number and consider the function f(x) = frac1x^2 + c x. Use Mathematica to calculate the Taylor polynomial of degree 5 for f(x) at x = 1. Use Mathematica to plot f(x) for 0 le x le 2, as well as the Taylor polynomials of degrees 1, 2 and 3 for f(x) at x = 1. [4 marks] Remark: Approximately one-ninth of you should be pleasantly surprised by your Taylor series! (d) In the lectures, we deduced that the Taylor series for tan^-1(x) at x = 0 is given by x - x^3/3 + x^5/5 - x^7/7 + cdots + (-1)^n+1 fracx^2n-12n-1 + cdots Combining this equation with the fact that pi = 4 an^-1(1), we obtain a series for pi. Use Mathematica to calculate the 1000th partial sum of the series to ten decimal places. How many of those ten decimal places agree with the decimal expansion of pi? [2 marks]$

Added by Jessica R.

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