The image presents the general term of a series:

$u_n = \frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right), \quad n \geq 1$

Key Details:

1. General Formula: The term is expressed as a fraction dependent on $n$ and is reformulated using partial fractions: $u_n = \frac{1}{n(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right).$

2. Partial Fraction Decomposition: The formula shows the decomposition of the term $\frac{1}{n(n+2)}$ into a simpler telescoping series.

3. Conditions: The term is valid for $n \geq 1$.

This is likely leading to telescoping summation for simplifying the series summation process.

Do you want to:

• Explore how to sum this series?
• Get additional details on partial fraction decomposition?

Related Questions:

1. What is the importance of partial fraction decomposition in simplifying series?
2. How does the telescoping property help in evaluating series sums?
3. Can the convergence of this series be verified?
4. How does the condition $n \geq 1$ impact the summation process?
5. Can we derive a closed-form sum for $S_n = \sum_{k=1}^n u_k$?

Tip:

When dealing with telescoping series, always check for terms that cancel out to simplify calculations significantly.