To solve the sum of the given power series, we have:

$S(x) = \sum_{n=0}^{\infty} \frac{(n+1)(n+2)}{n!} x^n$

Let's break down and solve the series step-by-step.

1. Factorization of the Numerator:

The numerator $(n+1)(n+2)$ can be expanded as follows:

$(n+1)(n+2) = n^2 + 3n + 2$

Thus, the series becomes:

$S(x) = \sum_{n=0}^{\infty} \frac{n^2 + 3n + 2}{n!} x^n$

2. Splitting the Series:

We can split the series into three separate series:

$S(x) = \sum_{n=0}^{\infty} \frac{n^2}{n!} x^n + 3\sum_{n=0}^{\infty} \frac{n}{n!} x^n + 2\sum_{n=0}^{\infty} \frac{1}{n!} x^n$

3. Identifying Standard Series:

Each part can be identified with standard exponential series:

• For $\sum_{n=0}^{\infty} \frac{n^2}{n!} x^n$, we can use the fact that $n^2 = n(n-1) + n$ to rewrite it as $\sum_{n=0}^{\infty} \frac{n(n-1)}{n!} x^n + \sum_{n=0}^{\infty} \frac{n}{n!} x^n$. This will lead to two series involving $e^x$ and its derivative.
• $\sum_{n=0}^{\infty} \frac{n}{n!} x^n = x \frac{d}{dx}\sum_{n=0}^{\infty} \frac{x^n}{n!} = xe^x$
• $\sum_{n=0}^{\infty} \frac{1}{n!} x^n = e^x$

Let's compute these separately.

4. Computing Each Component:

• First Component: $\sum_{n=0}^{\infty} \frac{n^2}{n!} x^n$

$\sum_{n=0}^{\infty} \frac{n^2}{n!} x^n = \sum_{n=0}^{\infty} \left(\frac{n(n-1)}{n!} + \frac{n}{n!}\right) x^n$

$= \sum_{n=0}^{\infty} \frac{x^n}{(n-2)!} + \sum_{n=0}^{\infty} \frac{x^n}{(n-1)!} = x^2e^x + xe^x$

• Second Component: $3\sum_{n=0}^{\infty} \frac{n}{n!} x^n = 3xe^x$

• Third Component: $2\sum_{n=0}^{\infty} \frac{1}{n!} x^n = 2e^x$

5. Combining the Components:

$S(x) = (x^2 e^x + x e^x) + 3(x e^x) + 2 e^x$

$S(x) = x^2 e^x + 4 x e^x + 2 e^x$

Thus, the sum of the series is:

$S(x) = e^x(x^2 + 4x + 2)$

This represents the closed-form of the sum of the power series.

Would you like a detailed breakdown of any specific part, or do you have any other questions? Here are some example questions you might find interesting:

1. How can the series be split into standard forms?
2. What techniques are useful in manipulating series like these?
3. How does the exponential function relate to power series?
4. What other forms can power series take?
5. How can differentiation help in solving power series problems?

Tip: When working with power series, identifying standard series like the exponential series can simplify the problem significantly.