ln(x+2) written as a Maclaurin series, using 1/1-x

Question

Find the Maclaurin series for ln(x+2) using the geo-series x^n (which is equal 1/(1+x)).

Answer 1

d/dx ln(x+2) = 1/(x+2) = ∑_n=0^∞ xⁿ(-1)ⁿ / 2ⁿ⁺¹

Integrating both sides

ln(x+2) = ∑_n=0^∞ xⁿ⁺¹(-1)ⁿ / ((n+1)2ⁿ⁺¹) + C

When x=0, C = ln(2)

ln(x+2) = ln(2) + ∑_n=0^∞ xⁿ⁺¹(-1)ⁿ / ((n+1)2ⁿ⁺¹⁾

which can proved to be equal to

ln(x+2) = ln(2)+((1/2)/1!)x¹−((1/4)/2!)x²+((1/4)/3!)x³− ...