Question

Answer the following:

Find the coefficient of x⁶ in the expansion of e^2x using series expansion

Answer 1

e^x = `1 + x/(1!) + x^2/(2!) + x^3/(3!) + x^4/(4!) + x^5/(5!) + x^6/(6!) + ...`

∴ e^2x = `1 + ((2x))/(1!) + (2x)^2/(2!) + (2x)^3/(3!) + (2x)^4/(4!) + (2x)^5/(5!) + (2x)^6/(6!)+ ...`

`=1 + 2/(1!)x + 2^2/(2!)x^2 + 2^3/(3!)x^3 + 2^4/(4!)x^4 + 2^5/(5!)x^5 + 2^6/(6!)x^6 + ...`

∴ coefficient of x⁶ = `2^6/(6!)`

= `(2 xx 2 xx 2 xx 2 xx 2 xx 2)/(6 xx 5 xx 4 xx 3 xx 2 xx 1)`

= `4/45`