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EXAMPLE 6Finding the Maclaurin Expansion for f(x)=excosx

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!+)(1x22!+x44!)

Then excosx=1(1x22!+x44!)+x(1x22!+x44!)+x22!(1x22!+x44!)+x33!(1x22!+x44!)+x44!(1x22!+x44!)+x55!(1x22!+x44!)+=(1x22+x424)+(xx32+x524)+(x22x44)+(x36x512)+x424+x5120+=1+x+(12+12)x2+(12+16)x3+(12414+124)x4+(124112+1120)x5+=1+x13x316x4130x5+