Radius of convergence of $1 + \frac {x^1} {1} + \frac {x^2} {2} + \frac {x^3} {3} + \dots$

Question

Can someone please help me to compute the radius of convergence of

$$1 + \frac {x^1} {1} + \frac {x^2} {2} + \frac {x^3} {3} + \dots ?$$

Answer 1

your series is $\sum a_n x^n$

with $a_n=\frac{1}{n}>0$

$\lim_{n \to +\infty} \frac{a_{n+1}}{a_n}=$

1.

which is the inverse of convergence radius.

your radius is $1$

Answer 2

Hint: $R=\lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|=\lim_{n \to \infty} \left|\frac{1/n}{1/(n+1)}\right|=\lim_{n \to \infty} \left|\frac{n+1}{n}\right|=1$