Assignment:
Q1.) Given that 1-x+x^2+...+(-x)^n is a power series representation for 1/(1+x), find a power series representation for (x^3)/(1+x^2).
Q2.) The Maclaurin series for f(x) is:
1+2x+((3x^2)/(2))+((4x^3)/(6))+...+(((n+1)x^n)/(n!))+...
Let h(x)= (the integral from 0 to x) f(t) dt. Write the Maclaurin series for h(x).
Q3.) The polynomial 1+7x+21x^2 is used to approximate f(x)=(1+x)^7 on the interval
-0.01 ≤ x ≤ 0.01.
a.) Use the Remainder Estimation Theorem to estimate the maximum absolute error.
b.) Use a graphical method to find the actual maximum absolute error.
Q4.) Use Euler's Formula to write (i/2)(e^(3iΘ)-e^(-3iΘ) as a trigonometric function of Θ.
a.) sin3Θ c.) cos3Θ e.) 2sin3Θ
b.) 2cos3Θ d.) -sin3Θ
Provide complete and step by step solution for the question and show calculations and use formulas.