# Question #59b21

##### 1 Answer

Nov 29, 2016

#### Explanation:

Let

We could form the TS from first principles, but it is easier to do so from the known TS for e^x

# y=2x{e^x}-2 #

# :. y=2x{1+x+x^2/(2!)+x^3/(3!)+x^4/(4!) + ...}-2 #

# :. y={2x+2x^2+(2x^3)/(2!)+(2x^4)/(3!)+(2x^5)/(4!) + ...}-2 #

# :. y=-2+2x+2x^2+(2x^3)/(2!)+(2x^4)/(3!)+(2x^5)/(4!) + ... #

Incidentally the series

If you meant

# y=2x(e^xe^-2) =2/e^2 xe^x#

# :. y=2/e^2x{1+x+x^2/(2!)+x^3/(3!)+x^4/(4!) + ...} #

# :. y=2/e^2{x+x^2+x^3/(2!)+x^4/(3!)+x^5/(4!) + ...} #

# :. y=2/e^2x+2/e^2x^2+2/e^2x^3/(2!)+2/e^2x^4/(3!)+2/e^2x^5/(4!) + ... #