Exercises 8.8 use the Taylor series given in Key Idea 32 to verify the given iden????ty. \(f(x)\quad =\quad sin(2x+3)\\\)
\(f^{ }\left( 0 \right) =\quad sin(3)\\\)
\(f^{ 1 }\left( x \right) \quad =\quad 2cos(2x+3),\quad f^{ 1 }\left( 0 \right) \quad =\quad 2cos(3)\\\)
\(f^{ 2 }\left( x \right) \quad =\quad -4sin(2x+3),\quad f^{ 2 }\left( 0 \right) \quad =\quad -4sin(3)\\\)
\(f^{ 3 }\left( x \right) \quad =\quad -8cos(2x+3),\quad f^{ 3 }\left( 0 \right) \quad =\quad -8cos(3)\\\)
\(f\left( x \right) \quad =\sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ 2 }^{ 2n }sin(3) } \frac { { x }^{ 2n } }{ (2n)! } \quad +\quad \sum _{ n=0 }^{ \infty }{ { (-1) }^{ n-1 } } { 2 }^{ n+1 }cos(3)\frac { { x }^{ 2n+1 } }{ (2n+1)! }\\\)