Chapter 9 - Power Series - Review Exercises - Page 705: 33

\[\begin{align} & \left. a \right)-\left( x-\frac{\pi }{2} \right)+\frac{1}{6}{{\left( x-\right)}^{3}}-\frac{1}{120}{{\left( x-\frac{\pi }{2} \right)}^{5}} \\ & \left. b \right)\sum\limits_{k=0}^{\infty }{\frac{{{\left( -1 \right)}^{k+1}}}{\left( 2k+1 \right)!}{{\left( x-\frac{\pi }{2} \right)}^{2k+1}}} \\ \end{align}\]

\[\begin{align} & f\left( x \right)=\cos x,\text{ }a=\frac{\pi }{2} \\ & \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{First}\text{, we calculate some derivatives and their value at }x=\pi \\ & f\left( x \right)=\cos x\to f\left( \frac{\pi }{2} \right)=\cos \left( \frac{\pi }{2} \right)=0 \\ & f'\left( x \right)=-\sin x\to f'\left( \frac{\pi }{2} \right)=-\sin \left( \frac{\pi }{2} \right)=-1 \\ & f''\left( x \right)=-\cos x\to f''\left( \frac{\pi }{2} \right)=-\cos \left( \frac{\pi }{2} \right)=0 \\ & f'''\left( x \right)=\sin x\to f'''\left( \frac{\pi }{2} \right)=\sin \left( \frac{\pi }{2} \right)=1 \\ & {{f}^{\left( 4 \right)}}\left( x \right)=\cos x\to {{f}^{\left( 4 \right)}}\left( \frac{\pi }{2} \right)=\cos \left( \frac{\pi }{2} \right)=0 \\ & {{f}^{\left( 5 \right)}}\left( x \right)=-\sin x\to {{f}^{\left( 5 \right)}}\left( \frac{\pi }{2} \right)=-\sin \left( \frac{\pi }{2} \right)=-1 \\ & \\ & \left. a \right)\text{ Substituting the previous result into the formula }\left( \mathbf{1} \right). \\ & \left( \text{Using the nonzero coefficients} \right) \\ & =\frac{\left( -1 \right)}{1!}{{\left( x-\frac{\pi }{2} \right)}^{1}}+\frac{\left( 1 \right)}{3!}{{\left( x-\frac{\pi }{2} \right)}^{3}}+\frac{\left( -1 \right)}{5!}{{\left( x-\frac{\pi }{2} \right)}^{5}} \\ & \text{Simplifying} \\ & =-\left( x-\frac{\pi }{2} \right)+\frac{1}{6}{{\left( x-\frac{\pi }{2} \right)}^{3}}-\frac{1}{120}{{\left( x-\frac{\pi }{2} \right)}^{5}} \\ & \\ & \left. b \right)\text{ Rewrite the power series} \\ & =\frac{{{\left( -1 \right)}^{0+1}}}{1!}\left( x-\frac{\pi }{2} \right)+\frac{{{\left( -1 \right)}^{1+1}}}{3!}{{\left( x-\frac{\pi }{2} \right)}^{3}}+\frac{{{\left( -1 \right)}^{2+1}}}{5!}{{\left( x-\frac{\pi }{2} \right)}^{5}} \\ & \text{Using summation notation}\text{, we obtain} \\ & \left( \text{Recall the formula for odd numbers }2n+1 \right) \\ & \sum\limits_{k=0}^{\infty }{\frac{{{\left( -1 \right)}^{k+1}}}{\left( 2k+1 \right)!}{{\left( x-\frac{\pi }{2} \right)}^{2k+1}}} \\ \end{align}\]