Find the Taylor series for $f(x) =\ln(1+x)$ Find the Taylor series for $f(x) = \ln (1+x)$ centered at $x = 0$ using the formula for Taylor Series.


$$\ln(1+x)=x-\dfrac {x^2}2+\dfrac {x^3}3-\dfrac {x^4}4+\&\text{c}=\sum\limits_{r=0}^{\infty}\dfrac {(-1)^{r}x^{r+1}}{r+1}$$

The Taylor Series for $f(x)$ at the point $x=a$ is
$$f(x)=f(a)+f^1(a)(x-a)+\dfrac {f^2(a)}{2!}(x-a)^2+\dfrac {f^3(a)}{3!}(x-a)^3+\cdots+\dfrac {f^r(a)}{r!}(x-a)^r$$
Since we want it at $x=0$, $a=0$ and simply plug them into the formula to get the expansion!

Note that another way is to start with the exponential theorem$$a^x=1+cx+\dfrac {c^2x^2}{2!}+\dfrac {c^3x^3}{3!}+\&\text{c}$$
Where $$c=(a-1)-\dfrac 12(a-1)^2+\dfrac 13(a-1)^3-\&\text c$$
And by setting $x=1,\ x=c$, we obtain$$a=e^c\implies c=\ln a$$
And therefore,$$\begin{align*} & \ln a=(a-1)-\dfrac 12(a-1)^2+\dfrac 13(a-1)^3-\&\text{c}\\ & \implies \ln(x+1)=x-\dfrac {x^2}2+\dfrac {x^3}3-\dfrac {x^4}4+\&\text c\end{align*}$$
