Find the Maclaurin series for ln(1+2x). For what values of x does the series converge?

pls solution :D

Question

Find the Maclaurin series for ln(1+2x). For what values of x does the series converge?

pls solution :D

tkhunny · Answer

Have you met this method?  (You may wish to reference "Uniform Convergence")

$\dfrac{d}{dx}ln(1+2x) = \dfrac{2}{1+2x} = 2(1 - 2x + (2x)^2 - (2x)^{3} + ...)$

Then,

$ln(1+2x) = \dfrac{2}{1}x - \dfrac{4}{2}x^{2} + \dfrac{8}{3}x^3 - \dfrac{16}{4}x^{4} + \dfrac{32}{5}x^{5} - ... +(-1)^{n-1}\dfrac{2^{n}}{n}x^{n} + ...$ 

Sometimes, either the derivative or the antiderivative is easier!

You have to do the convergence part!  Ratio test?

anonymous · Answer

thanks big help :D