To find the radius of convergence of the series $\sum_{n=0}^{\infty} (n+1)x^n$, we will use the ratio test, which states that the radius of convergence $R$ is given by:

$\frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

where $a_n$ is the $n$-th term of the series. In this case, the general term of the series is:

$a_n = (n+1)x^n$

Applying the ratio test:

$\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(n+2)x^{n+1}}{(n+1)x^n} \right| = \left| \frac{(n+2)x}{(n+1)} \right|$

Simplifying further:

$\left| \frac{a_{n+1}}{a_n} \right| = \left| x \cdot \frac{n+2}{n+1} \right|$

Now, let's find the limit as $n \to \infty$:

$\lim_{n \to \infty} \left| x \cdot \frac{n+2}{n+1} \right| = \left| x \cdot \lim_{n \to \infty} \frac{n+2}{n+1} \right|$

Since $\frac{n+2}{n+1} \to 1$ as $n \to \infty$, we have:

$\frac{1}{R} = |x|$

Thus, the radius of convergence $R$ is:

$R = \frac{1}{|x|} = 1$

Therefore, the radius of convergence of the series is:

$R = 1$

Would you like further details or have any questions? Here are some related topics you might find interesting:

1. How to determine the interval of convergence of a series?
2. What is the ratio test and how is it applied to series?
3. How does the root test compare to the ratio test for determining convergence?
4. What happens if the ratio test result is inconclusive?
5. How to handle series with complex terms in finding the radius of convergence?

Tip: When using the ratio test, always simplify the expression as much as possible before taking the limit to avoid errors in finding the radius of convergence.