Finding the co-efficients of this power series I am required to find the co-efficients of this power series: $2x\ln(1+2x)$
I approached the problem by considering the $\ln(1+2x)$ part as the integral of $2/(1+2x)$ and applied the geometric series to this. Factoring the $2$ on the numerator outside this expression gives us $4x$ on the outside (after multiplying it with $2x$). 
I need to get the first 5 nonzero co-efficients, and what I got is: $0, 4, -8, 16$ and $-32$. Only $0$ is right, according to the online portal. 
How do I approach this problem?

$\log(1+2x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(2x)^n}{n}$ implies that 
$$2x\log(1+2x)=\sum_{n=1}^{\infty} \frac{(-2x)^{n+1} }{n}=\sum_{n=2}^{\infty} \frac{(-2x)^{n} }{n-1}$$
