The provided image contains three mathematical expressions involving summations and limits. Let's break down each component step-by-step:

1. $F(n)$

The function $F(n)$ is given by a double summation:

$F(n) = \sum_{m=1}^{n} \left( \binom{n}{m} m! \sum_{k=0}^{m} \frac{(-1)^k}{k!} \right) + 1$

• $\binom{n}{m}$ represents the binomial coefficient, which counts the ways to choose $m$ elements from a set of $n$ elements.
• The inner sum involves $(-1)^k$ divided by $k!$, which is part of the exponential series expansion for $e^{-1}$.
• The outer sum runs from $m = 1$ to $n$, multiplying the binomial coefficient $\binom{n}{m}$ by $m!$.

2. $G(n)$

The second expression defines $G(n)$:

$G(n) = \sum_{k=0}^{n} \binom{2n}{2k} \binom{2n - 2k}{n - k} \binom{2k}{k}$

• Here, $\binom{2n}{2k}$, $\binom{2n - 2k}{n - k}$, and $\binom{2k}{k}$ are binomial coefficients.
• The sum runs from $k = 0$ to $n$, making this expression another combinatorial identity that is likely related to counting lattice paths or combinatorial structures.

3. Limit expression

The third part of the expression is a limit:

$\lim_{n \to \infty} \left( \frac{\sqrt{G(n) F(n)}}{n^n} \right)^{\frac{1}{n}} = ?$

This is the limit of the expression involving $G(n)$ and $F(n)$ as $n \to \infty$, normalized by $n^n$. The $n^n$ term suggests that we are looking for the asymptotic behavior of $F(n)$ and $G(n)$ as $n$ grows large.

To solve:

• We would need to find the asymptotic behavior of $F(n)$ and $G(n)$, potentially using approximations for large $n$.
• Then, apply these approximations to evaluate the limit.

Would you like a detailed calculation of this limit step-by-step or further explanation on specific components of these sums?

Relative Questions:

1. How do we evaluate asymptotic behavior of summations as $n \to \infty$?
2. What are common techniques for approximating large factorials and binomial coefficients?
3. How does Stirling's approximation help in analyzing large factorials in summations?
4. Can we simplify combinatorial sums like $G(n)$ using known identities?
5. What is the significance of the $n^n$ normalization in the limit?

Tip:

When dealing with large summations involving factorials and binomial coefficients, asymptotic approximations like Stirling's approximation can simplify the analysis significantly.