Determine a Taylor Series about \(x = 0\) for the following integral.
\[\int{{\frac{{\cos \left( x \right) - 1}}{x}\,dx}}\]
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 \(\left[ { - \pi ,2\pi } \right]\).
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 \(\left[ { - 4,0} \right]\).
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 \(\left[ {1,5} \right]\).
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 \(\left[ { - 2,4} \right]\).