<s>[INST] For which $x$ does the series $∑_{n=1}^∞(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n})\, x^n$ converge? Determine for what value of $x$ the series converges
$$\sum_{n=1}^\infty \left(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n}\right) x^n $$
Observe that 
$∑_{n=1}^∞(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n}) x^n =∑_{n=1}^∞ (∑_{n=1}^∞\frac{1}{n}) x^n$
Can I consider $(\sum_{n=1}^\infty\frac{1}{n})$ as $a_n$ and use the ratio test here or I must use the Hadamard?
 [/INST] Note that 
$$
1\le 1+\frac{1}{2}+\cdots+\frac{1}{n}\le \log (n+1),
$$
and hence, using for example the comparison test, the radius of convergence of the series 
$$\sum_{n=1}^\infty \left(1+\frac{1}{2}+ \frac{1}{3}+⋯+\frac{1}{n}\right) x^n $$
is $r=1$,  since the radius of convergence of both series
$$
\sum_{n=0}^\infty x^n, \quad \sum_{n=0}^\infty \log n\, x^n
$$
is equal to $1$.
</s>