Find the first five terms of the Maclaurin expansion for f(x)=excosx.
Solution The Maclaurin expansion for f(x)= excosx is obtained by multiplying the Maclaurin expansion for ex by the Maclaurin expansion for cosx. That is, excosx=(1+x+x22!+x33!+x44!+x55!+⋯)(1−x22!+x44!−⋯)
Then excosx=1(1−x22!+x44!−⋯)+x(1−x22!+x44!−⋯)+x22!(1−x22!+x44!−⋯)+x33!(1−x22!+x44!−⋯)+x44!(1−x22!+x44!−⋯)+x55!(1−x22!+x44!−⋯)+⋯=(1−x22+x424)+(x−x32+x524)+(x22−x44)+(x36−x512)+x424+x5120+⋯=1+x+(−12+12)x2+(−12+16)x3+(124−14+124)x4+(124−112+1120)x5+⋯=1+x−13x3−16x4−130x5+⋯