Write the Maclaurin’s series for f(x)=sin(3x)+e^x up to the third order
To simplify this question, it is possible to expand the two elements of the function and then add the two expansions together.First, the expansion of sin(3x) around the origin is sin(30)+d/dx sin(3x)+d²/dx² sin(3x)+d³/dx³ sin(3x)=sin(30)+3cos(30)x-33sin(30)x²/2!-333cos(3*0)*x³/3!+…=0+3x+0x²-27x³/3!+…=0+3x+0x²-9x³/2+… (1)Then, the expansion of e^x is trivial as 1+x+x²+x³… (2) and can be added to our previous result (1), obtaining the final result: f(x)=1+4x+x²/2-25x³/6+…
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