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Section 4-17 : Applications of Series

1. Determine a Taylor Series about $$x = 0$$ for the following integral. $\int{{\frac{{\cos \left( x \right) - 1}}{x}\,dx}}$
2. Write down $${T_2}\left( x \right)$$, $${T_4}\left( x \right)$$ and $${T_6}\left( x \right)$$ for the Taylor Series of $$f\left( x \right) = \sin \left( x \right)$$ about $$\displaystyle x = \frac{{3\pi }}{2}$$. Graph all three of the Taylor polynomials and $$f\left( x \right)$$ on the same graph for the interval $$\left[ { - \pi ,2\pi } \right]$$.
3. Write down $${T_2}\left( x \right)$$, $${T_3}\left( x \right)$$ and $${T_4}\left( x \right)$$ for the Taylor Series of $$f\left( x \right) = \ln \left( {1 - x} \right)$$ about $$x = - 2$$. Graph all three of the Taylor polynomials and $$f\left( x \right)$$ on the same graph for the interval $$\left[ { - 4,0} \right]$$.
4. Write down $${T_1}\left( x \right)$$, $${T_3}\left( x \right)$$ and $${T_5}\left( x \right)$$ for the Taylor Series of $$f\left( x \right) = \frac{1}{{{{\left( {6 - x} \right)}^7}}}$$ about $$x = 4$$. Graph all three of the Taylor polynomials and $$f\left( x \right)$$ on the same graph for the interval $$\left[ {1,5} \right]$$.
5. Write down $${T_2}\left( x \right)$$, $${T_4}\left( x \right)$$ and $${T_6}\left( x \right)$$ for the Taylor Series of $$f\left( x \right) = \sqrt {2 + x}$$ about $$x = 1$$. Graph all three of the Taylor polynomials and $$f\left( x \right)$$ on the same graph for the interval $$\left[ { - 2,4} \right]$$.