How do I work out the validity for a Maclaurin (power) series?

Question

I cannot find the answer to this anywhere so I have decided to make a question.

Given a Maclaurin series for a function, how can I quickly work out what the validity is for it?

For example, $\arctan(\frac{1}{2}x-2)$ OR $\ln(3x+5)$

Is there a quick way to work out the validity?

Answer 1

You know $\ln(1 +x)$ is valid for $|x| \leq 1$ so $x \mapsto f(x)$ means the validity is restricted to $|f(x)| \leq 1$.

Write $\ln(3x+5) = \ln 5 + \ln \left(1 + \frac{3x}{5}\right)$ then the validity is $\left|\frac{3x}{5}\right| \leq 1 \iff |x|\leq \frac{5}{3}$ which is your radius of convergence.

You can do something similar for $\arctan \left(\frac{x}{2} - 2\right)$.