comparison test ∑∞𝑛=1𝑛!(𝑛+2)!




The **comparison test** helps determine the convergence or divergence of a series by comparing it to another series that is known to converge or diverge. To use the comparison test on

\[
\sum_{n=1}^{\infty} \frac{n!}{(n+2)!},
\]

we need to compare this series to a simpler one whose convergence properties are well-known. Here’s a step-by-step analysis:

1. **Simplify the Terms**: Start by simplifying the expression \( \frac{n!}{(n+2)!} \).
   - Note that \((n+2)! = (n+2)(n+1)n!\), so we can rewrite \(\frac{n!}{(n+2)!}\) as:
     \[
     \frac{n!}{(n+2)(n+1)n!} = \frac{1}{(n+2)(n+1)}.
     \]

2. **Form of the Simplified Series**: This means the series becomes:
   \[
   \sum_{n=1}^{\infty} \frac{1}{(n+2)(n+1)}.
   \]
   
3. **Comparison with a Known Series**: Now, let’s compare \(\frac{1}{(n+2)(n+1)}\) to a simpler form that we can analyze. Notice that:
   \[
   \frac{1}{(n+2)(n+1)} < \frac{1}{n^2}.
   \]
   
   Since the series \(\sum \frac{1}{n^2}\) is a convergent \(p\)-series (with \(p = 2 > 1\)), and \(\frac{1}{(n+2)(n+1)}\) is smaller than \(\frac{1}{n^2}\) for large \(n\), we can conclude that our original series \(\sum \frac{n!}{(n+2)!}\) converges by the **comparison test**.

### Conclusion:
\[
\sum_{n=1}^{\infty} \frac{n!}{(n+2)!} \text{ converges}.
\]

Would you like a more detailed explanation or have any other questions?

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Here are some related questions to deepen understanding:

1. What are some other convergence tests that can be used on factorial-based series?
2. How does the **limit comparison test** differ from the **comparison test**, and can it be applied here?
3. Could the **ratio test** also determine convergence for this series?
4. What are \(p\)-series, and why are they useful in convergence analysis?
5. How can factorial simplification techniques assist in evaluating series convergence?

**Tip:** For factorial expressions in series, try simplifying the ratio of consecutive terms to make convergence easier to analyze.

comparison test ∑∞𝑛=1𝑛!(𝑛+2)!

Explore how to determine the convergence of the series ∑ n!/(n+2)! using the Comparison Test. This solution breaks down factorial simplification and compares the result to a known convergent p-series, providing insights for undergraduate-level students studying series convergence.

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