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The question involves analyzing the convergence of the series \( \sum_{n=1}^{\infty} \frac{n^2 + 2}{n!} \) using various methods:

### (a) **Ratio Test**:
The series is tested for convergence using the **Ratio Test**. The Ratio Test involves evaluating:
\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]
where \( a_n = \frac{n^2 + 2}{n!} \). For the series to converge, \( L \) must be less than 1.

Start by computing:
\[
a_n = \frac{n^2 + 2}{n!}, \quad a_{n+1} = \frac{(n+1)^2 + 2}{(n+1)!}
\]
The ratio becomes:
\[
\frac{a_{n+1}}{a_n} = \frac{(n+1)^2 + 2}{(n+1)!} \times \frac{n!}{n^2 + 2}
= \frac{(n+1)^2 + 2}{(n+1)(n^2 + 2)}
\]
We need to evaluate this limit as \( n \to \infty \). Since the factorial in the denominator grows rapidly, this limit should approach 0, which is less than 1. Therefore, by the Ratio Test, the series converges.

### (b) **Partial Sums**:
You are asked to find the partial sums \( S_n \) for \( n = 5, 10, 15, 20, 25 \). These partial sums can be found by calculating:
\[
S_n = \sum_{k=1}^{n} \frac{k^2 + 2}{k!}
\]
For example:
\[
S_5 = \frac{1^2 + 2}{1!} + \frac{2^2 + 2}{2!} + \frac{3^2 + 2}{3!} + \frac{4^2 + 2}{4!} + \frac{5^2 + 2}{5!}
= 3 + 3 + 4 + 1.67 + 0.73 = 12.4
\]
Continue this calculation to fill in the table for \( S_{10}, S_{15}, S_{20}, \) and \( S_{25} \).

### (c) **Graph of Partial Sums**:
The graphing utility should show the partial sums for the first 10 terms. From the graph images provided, the sums seem to converge as \( n \to 10 \). This is consistent with the behavior of convergent series, where the partial sums approach a horizontal asymptote.

### (d) **Estimate of the Sum**:
The sum of the series can be estimated using the table of partial sums, likely using the value at \( S_{25} \), as it gives a reasonable approximation of the total sum.

Would you like further details on any part of this? Here are some follow-up questions to deepen your understanding:

1. How is the Ratio Test generally applied to series?
2. Why does the factorial in the denominator lead to convergence in this series?
3. Can you explain the meaning of partial sums in the context of series?
4. How could the partial sums be computed using a calculator or software?
5. What can the graph of partial sums tell you about the nature of the series' convergence?

**Tip**: For large factorial terms in series, the Ratio Test often simplifies significantly since factorials grow much faster than polynomials.

The problem involves analyzing the convergence of the series sum_{n=1}^{∞} (n^2 + 2) / n! using the Ratio Test, computing partial sums, graphing the partial sums, and estimating the total sum.

This problem explores the convergence of the series sum_{n=1}^{∞} (n^2 + 2) / n! using the Ratio Test. It involves calculating partial sums, graphing the first 10 terms, and estimating the sum. This is suitable for college-level students studying series and sequences in calculus.

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