Maclaurin's series

Question

Expand e^{sin x} as a series of ascending power of x upto term containing x⁴.

Answer 1

e^x = 1 + x + x²/2 + x³/6 + x⁴/24^{+ ...}

 

sinx = 1 - x³/6 + x⁵/120 - ...

 

So, e^xsinx = (1 + x + x²/2 + x³/6 + x⁴/24 + ...)(1 - x³/6 + x⁵/120 - ...)

 

                = 1 - x³/6 + x - x⁴/6 + x²/2 + x³/6 + x⁴/24+...

 

                = 1 + x + x²/2 - x⁴/8 + ...