To determine if the series

$\sum_{n=2}^{\infty} \left(\frac{1}{n^2 - n}\right)$

converges by the comparison test, we can start by simplifying the general term and comparing it to a known convergent series.

Step 1: Simplify the Term

We can rewrite $\frac{1}{n^2 - n}$ as follows:

$\frac{1}{n^2 - n} = \frac{1}{n(n - 1)}$

Step 2: Partial Fraction Decomposition

We can break down $\frac{1}{n(n - 1)}$ using partial fraction decomposition:

$\frac{1}{n(n - 1)} = \frac{1}{n - 1} - \frac{1}{n}$

So, the series can be rewritten as:

$\sum_{n=2}^{\infty} \left(\frac{1}{n - 1} - \frac{1}{n}\right)$

This is a telescoping series, where most terms will cancel out, leaving us with only a few terms from the beginning of the series.

Step 3: Write Out the First Few Terms

Writing out the first few terms, we have:

$\left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \cdots$

In this telescoping series, all intermediate terms cancel, leaving only:

$1 - \lim_{n \to \infty} \frac{1}{n} = 1$

Conclusion

Since the partial sums converge to a finite value (1), the series $\sum_{n=2}^{\infty} \frac{1}{n^2 - n}$ converges.

The Comparison Test isn't directly used here because we transformed the series into a telescoping series. However, if we wanted to apply the comparison test, we could note that $\frac{1}{n(n-1)}$ is smaller than $\frac{1}{n^2}$, which converges (since the $p$-series $\sum \frac{1}{n^p}$ with $p > 1$ converges).

Would you like further details on telescoping series, or have any questions?

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Related Questions

1. How does partial fraction decomposition work for series convergence?
2. What is the telescoping series test and when is it useful?
3. How does the $p$-series test for convergence apply to comparison tests?
4. Can a series be convergent by both telescoping and comparison tests?
5. How does the limit comparison test differ from the direct comparison test?

Tip:

For series with expressions that factor easily, consider telescoping to simplify convergence testing.