Show All Notes Hide All Notes

Section 10.15 : Power Series and Functions

For problems 1 – 4 write the given function as a power series and give the interval of convergence.

1. \( \displaystyle f\left( x \right) = \frac{x}{{1 - 8x}}\)
2. \( \displaystyle f\left( x \right) = \frac{{ - 12{x^2}}}{{1 + 6{x^7}}}\)
3. \( \displaystyle f\left( x \right) = \frac{{{x^7}}}{{8 + {x^3}}}\)
4. \( \displaystyle f\left( x \right) = \frac{{\sqrt[5]{{{x^2}}}}}{{4 - 3{x^2}}}\)

For problems 5 & 6 give a power series representation for the derivative of the following function.

1. \( \displaystyle g\left( x \right) = \frac{{{x^{10}}}}{{2 - {x^2}}}\)
2. \( \displaystyle g\left( x \right) = \frac{{9{x^5}}}{{1 + 3{x^6}}}\)

For problems 7 & 8 give a power series representation for the integral of the following function.

1. \( \displaystyle h\left( x \right) = \frac{{7x}}{{3 - 6x}}\)
2. \( \displaystyle h\left( x \right) = \frac{{{x^4}}}{{2 + {x^9}}}\)