Chapter 9 - Power Series - 9.3 Taylor Series - 9.3 Exercises - Page 695: 27

\[\begin{align} & \left. a \right)2+2\left( \ln 2 \right)\left( x-1 \right)+{{\left( \ln 2 \right)}^{2}}{{\left( x-1 \right)}^{2}}+\frac{{{\left( \ln 2 \right)}^{3}}}{3}{{\left( x-1 \right)}^{3}} \\ & \left. b \right)\sum\limits_{k=0}^{\infty }{\frac{2{{\left( \ln 2 \right)}^{k}}{{\left( x-1 \right)}^{k}}}{k!}} \\ \end{align}\]

\[\begin{align} & f\left( x \right)={{2}^{x}},\text{ }a=1 \\ & \text{Using the definition of Taylor series for a function}\left( page\,685 \right) \\ & \sum\limits_{k=0}^{\infty }{\frac{{{f}^{\left( k \right)}}\left( a \right)}{k!}{{\left( x-a \right)}^{k}}}\text{ }\left( \mathbf{1} \right) \\ & =f\left( a \right)+f'\left( a \right)\left( x-a \right)+\frac{f''\left( a \right)}{2!}{{\left( x-a \right)}^{2}}+\cdots \\ & \text{Calculate some derivatives and their value at }x=1 \\ & f\left( x \right)={{2}^{x}}\to f\left( 1 \right)={{2}^{1}}=2 \\ & f'\left( x \right)={{2}^{x}}\left( \ln 2 \right)\to f\left( 1 \right)=2\left( \ln 2 \right) \\ & f''\left( x \right)={{2}^{x}}{{\left( \ln 2 \right)}^{2}}\to f\left( 1 \right)=2{{\left( \ln 2 \right)}^{2}} \\ & f''\left( x \right)={{2}^{x}}{{\left( \ln 2 \right)}^{3}}\to f\left( 1 \right)=2{{\left( \ln 2 \right)}^{3}} \\ & \\ & \left. a \right)\text{ Substituting the previous result into the formula }\left( \mathbf{1} \right). \\ & \left( \text{Using the nonzero coefficients} \right) \\ & =\frac{2}{0!}{{\left( x-1 \right)}^{0}}+\frac{2\left( \ln 2 \right)}{1!}\left( x-1 \right)+\frac{2{{\left( \ln 2 \right)}^{2}}}{2!}{{\left( x-1 \right)}^{2}}+\frac{2{{\left( \ln 2 \right)}^{3}}}{3!}{{\left( x-1 \right)}^{3}} \\ & \text{Simplifying} \\ & =2+2\left( \ln 2 \right)\left( x-1 \right)+{{\left( \ln 2 \right)}^{2}}{{\left( x-1 \right)}^{2}}+\frac{{{\left( \ln 2 \right)}^{3}}}{3}{{\left( x-1 \right)}^{3}} \\ & \\ & \left. b \right)\text{ Rewrite the power series} \\ & =\frac{2}{0!}{{\left( x-1 \right)}^{0}}+\frac{2\left( \ln 2 \right)}{1!}\left( x-1 \right)+\frac{2{{\left( \ln 2 \right)}^{2}}}{2!}{{\left( x-1 \right)}^{2}}+\frac{2{{\left( \ln 2 \right)}^{3}}}{3!}{{\left( x-1 \right)}^{3}} \\ & =2\left[ \frac{{{\left( \ln 2 \right)}^{0}}{{\left( x-1 \right)}^{0}}}{0!}+\frac{\left( \ln 2 \right)\left( x-1 \right)}{1!}+\frac{{{\left( \ln 2 \right)}^{2}}{{\left( x-1 \right)}^{2}}}{2!} \right] \\ & +2\left[ \frac{{{\left( \ln 2 \right)}^{3}}{{\left( x-1 \right)}^{3}}}{3!} \right] \\ & \text{Using summation notation}\text{, we obtain} \\ & =\sum\limits_{k=0}^{\infty }{\frac{2{{\left( \ln 2 \right)}^{k}}{{\left( x-1 \right)}^{k}}}{k!}} \\ \end{align}\]