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### Section 4-15 : 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. $$\displaystyle f\left( x \right) = \frac{{\sqrt{{{x^2}}}}}{{4 - 3{x^2}}}$$

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. $$\displaystyle h\left( x \right) = \frac{{{x^4}}}{{2 + {x^9}}}$$