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The series presented is:

\[
\sum_{k=4}^{\infty} \frac{5}{k^2 - k}
\]

We can start by simplifying the term inside the summation.

\[
\frac{5}{k^2 - k} = \frac{5}{k(k-1)}
\]

This expression can be decomposed using **partial fraction decomposition** as follows:

\[
\frac{5}{k(k-1)} = \frac{A}{k-1} + \frac{B}{k}
\]

To find \( A \) and \( B \), we solve the equation:

\[
\frac{5}{k(k-1)} = \frac{A}{k-1} + \frac{B}{k} 
\]

Multiplying both sides by \( k(k-1) \) gives:

\[
5 = A \cdot k + B \cdot (k-1)
\]

Expanding the right-hand side:

\[
5 = A k + B k - B
\]
\[
5 = (A + B)k - B
\]

Now, by comparing coefficients of \( k \) and the constant terms, we get the following system of equations:

1. \( A + B = 0 \)
2. \( -B = 5 \)

From equation (2), we have \( B = -5 \). Substituting into equation (1), we get:

\[
A - 5 = 0 \implies A = 5
\]

So the partial fraction decomposition is:

\[
\frac{5}{k(k-1)} = \frac{5}{k-1} - \frac{5}{k}
\]

Now, we substitute this into the original summation:

\[
\sum_{k=4}^{\infty} \frac{5}{k(k-1)} = \sum_{k=4}^{\infty} \left( \frac{5}{k-1} - \frac{5}{k} \right)
\]

This is a **telescoping series**, where most terms cancel out. Let's expand the first few terms to see the cancellation:

\[
\left( \frac{5}{3} - \frac{5}{4} \right) + \left( \frac{5}{4} - \frac{5}{5} \right) + \left( \frac{5}{5} - \frac{5}{6} \right) + \cdots
\]

All intermediate terms cancel out, and we're left with:

\[
\frac{5}{3}
\]

Thus, the sum of the series is:

\[
\boxed{\frac{5}{3}}
\]

---

Do you have any questions or would you like more details on any step?

Here are 5 related questions to expand on this:

1. What is a telescoping series, and how does it simplify the summation process?
2. Can you provide another example where partial fraction decomposition is used in summing a series?
3. How would you approach the series if the lower bound of the summation was different?
4. Why do most terms cancel in a telescoping series, and can you give an example where they do not?
5. How can this method of solving be applied to summing infinite series with higher powers in the denominator?

**Tip:** When dealing with rational expressions in series, always check if partial fraction decomposition can simplify the problem—it often turns a complex sum into a telescoping series!

Find the sum of the series: ∑ (from k=4 to ∞) 5 / (k^2 - k)

Learn how to find the sum of the infinite series ∑ (from k=4 to ∞) 5 / (k^2 - k) using partial fraction decomposition and the telescoping series method. Understand key concepts like series simplification and cancellation for students in Grades 11-12.

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