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Section 10.17 : Applications of Series

1. Determine a Taylor Series about \(x = 0\) for the following integral. \[\int{{\frac{{{{\bf{e}}^x} - 1}}{x}\,dx}}\] Solution
2. Write down \({T_2}\left( x \right)\), \({T_3}\left( x \right)\) and \({T_4}\left( x \right)\) for the Taylor Series of \(f\left( x \right) = {{\bf{e}}^{ - 6x}}\) about \(x = - 4\). Graph all three of the Taylor polynomials and \(f\left( x \right)\) on the same graph for the interval \(\left[ { - 8, - 2} \right]\). Solution
3. Write down \({T_3}\left( x \right)\), \({T_4}\left( x \right)\) and \({T_5}\left( x \right)\) for the Taylor Series of \(f\left( x \right) = \ln \left( {3 + 4x} \right)\) about \(x = 0\). Graph all three of the Taylor polynomials and \(f\left( x \right)\) on the same graph for the interval \(\left[ {\displaystyle - \frac{1}{2},2} \right]\). Solution