# Power Series, Maclaurin Series, Remainder Estimation Theorem and Euler's Formula

1.) 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).

2.) 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).

3.) 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.

4.) 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

https://brainmass.com/math/real-analysis/remainder-estimation-theorem-eulers-formula-123648

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Series

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1.) 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).

Solution. As , we have

Hence,

2.) The Maclaurin series for f(x) is:

...

#### Solution Summary

Power Series, Maclaurin Series, Remainder Estimation Theorem and Euler's Formula are investigated.