To find a function for the infinite series:

$S = \sum_{n=1}^{\infty} (\ln x)^n$

where $x > 0$ and $\ln x$ represents the natural logarithm of $x$, we recognize that this series has the form of a geometric series. A geometric series has the general form:

$\sum_{n=1}^{\infty} r^n = \frac{r}{1 - r}, \quad \text{for } |r| < 1.$

Here, $r = \ln x$, and the series starts at $n = 1$, not $n = 0$, so we can directly apply this formula:

$S = \frac{\ln x}{1 - \ln x}, \quad \text{for } |\ln x| < 1.$

Conditions:

The condition $|\ln x| < 1$ implies that $x$ must lie in the range $e^{-1} < x < e^1$, or approximately $0.3679 < x < 2.718$.

Thus, the sum of the infinite series is:

$S = \frac{\ln x}{1 - \ln x}, \quad \text{for } e^{-1} < x < e.$

Would you like further details or have any questions about the steps?

Here are 5 related questions:

1. What is the geometric series and how is it derived?
2. Can this approach be used for other logarithmic-based series?
3. What happens to the series when $x = 1$?
4. How do you apply the radius of convergence to geometric series involving $\ln x$?
5. How would the series change if it started at $n = 0$ instead?

Tip: Whenever dealing with an infinite series, always check the convergence criteria to avoid undefined or divergent sums!