The original question asked for which $b > 0$, the series of function $\sum_{n = 0}^{\infty} \frac{x^n} {n}$ is uniformly convergent on $(0, b)$.
When $0 < b < 1$, we know $\sum_{n = 0}^{\infty} \frac{b^n} {n}$ converges by root test. So by Weierstrass's M-test, $\sum_{n = 0}^{\infty} \frac{x^n} {n}$ is uniformly convergent on $(0, b)$.
On the other hand, when $b > 1$, $\sum_{n = 0}^{\infty} \frac{x^n} {n}$ is not even pointwisely convergent at $1$ since the harmonic series diverges.
But I am stuck on $b = 1$, how should I proceed?
EDIT-1:
The following is from Bartle's Introduction to Real Analysis:
9.4.6 Weierstrass M-Test Let $(M_n)$ be a sequence of positive real numbers such that $|f_n(x)| \le M_n$ for $x \in D, n \in \mathbb{N}$. If the series $\sum M_n$ is convergent, then $\sum f_n$ is uniformly convergent on $D$.