The given problem is to evaluate the infinite series:

$\sum_{n=1}^\infty \frac{x^n}{n}.$

Solution:

This series is related to the natural logarithm function. Specifically, the series can be expressed as:

$\sum_{n=1}^\infty \frac{x^n}{n} = -\ln(1 - x), \quad \text{for } |x| < 1.$

Explanation:

1. General Series Form: The formula for the natural logarithm is derived from the Taylor expansion: $\ln(1 - x) = -\sum_{n=1}^\infty \frac{x^n}{n}, \quad \text{for } |x| < 1.$ Negating both sides gives: $\sum_{n=1}^\infty \frac{x^n}{n} = -\ln(1 - x).$

2. Convergence: This series converges only when $|x| < 1$. For $|x| \geq 1$, it diverges.

Thus, the result of the series is: $\boxed{-\ln(1 - x), \quad |x| < 1.}$

Would you like a deeper explanation or assistance with related problems?

Related Questions:

1. How does the convergence of this series change if $|x| = 1$?
2. What is the Taylor series expansion of $-\ln(1 - x)$?
3. Can this series be rewritten in terms of the $n$-th partial sum?
4. What happens if $x > 1$? Does the series diverge, and why?
5. How does this series relate to the geometric series $\sum_{n=1}^\infty x^n$?

Tip:

When working with series expansions, always verify the radius of convergence to ensure the results are valid!