The image shows a series representation which involves a summation and logarithmic terms. Here's a breakdown of the mathematical expression they've written:

The series is written as:

$\sum_{n=1}^{\infty} \frac{x^n}{n+k} = -\frac{1}{x^k} \left( x + \frac{x^2}{2} + \dots + \frac{x^k}{k} + \log(1-x) \right)$

Where:

• The summation starts from $n = 1$ and goes to infinity, involving terms of the form $\frac{x^n}{n+k}$.
• The right-hand side is a closed form that involves several components:
  • $-\frac{1}{x^k}$ is a multiplying factor.
  • Inside the parentheses, we have a series that involves the powers of $x$ up to $\frac{x^k}{k}$, followed by the logarithmic term $\log(1-x)$.

The expression applies when:

• $k$ is a positive integer.
• $|x| < 1$.

Would you like more details about how this series is derived, or do you have specific questions?

Here are 5 related questions you might consider:

1. What is the logarithmic series, and how is it related to the sum?
2. How does the convergence of the series depend on $|x| < 1$?
3. How is this series used in approximations for small values of $x$?
4. Can you derive the $\log(1-x)$ term from the series expansion?
5. What happens when $k = 1$ or other specific values of $k$?

Tip: Series expansions like this are helpful when approximating functions for small values of variables, especially when $|x| < 1$.