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

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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).

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

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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.

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