I've calculated the first $4$ derivatives of $f(x)$ and evaluated each at $x=0$. I now have the following maclaurin series:
$$ 0 + 5x – \frac{25}{2!}x^2 + \frac{250}{3!}x^3 – \frac{3750}{4!}x^4 $$
I am awful at recognizing patterns, and need to come up with a series for this from $n=1$ to infinity. Additionally, if anyone has any tips for recognizing patterns like this, let me know, because it's something I need to be better at.
Best Answer
If you know that $\ln(1+x) =\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1}x^k}{k} $, just substitute $5x$ for $x$ to get $\ln(1+5x) =\sum_{k=1}^{\infty} \dfrac{(-1)^{k-1}(5x)^k}{k} =\sum_{k=1}^{\infty} x^k\dfrac{(-1)^{k-1}5^k}{k} $.