expand ln (x + 2)^3 until 3rd terms

Question

its about power series use a formula (1+x) = x^2-(x^2/2)+(x^3/3)-.......but for use this formula.i must factor the equation..i dont know how to factorize

Answer 1

Consider 1/(1+x)=(1+x)^-1=1-x+x²-x³+x⁴-... Now integrate wrt x: x-x²/2+x³/3-x⁴/4+x⁵/5-...

But the integral of 1/(1+x) is ln(1+x), so ln(1+x) expands to x-x²/2+x³/3-x⁴/4+x⁵/5-...

Note that this formula is given in the instructions within the question; I've just showed where the formula came from.

ln(x+2)³=3ln(x+2)=3ln(2+x)=3ln[2(1+x/2)]=3ln(2)+3ln(1+x/2).

If we replace x by x/2 in the expansion of ln(1+x) we get:

x/2-x²/8+x³/24-x⁴/64+x⁵/160-...

From this we can deduce the expansion of ln(x+2)³=

3ln(2)+3x/2-3x²/8+x³/8-3x⁴/64+3x⁵/160-...

Expansion to the third term is 3ln(2)+3x/2-3x²/8.