https://file.aiquickdraw.com/mathbot/file/1729335905402j5ko8ji8.jpg

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 \( |x| < 1 \).

Explore how to write the summation series with logarithmic terms for x and k. This problem breaks down the series expansion of \sum_{n=1}^{\infty} \frac{x^n}{n+k}, including the logarithmic function log(1-x) and its relationship to the summation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation