Question

Show that \(\lim _{k \rightarrow \infty} \int_{0}^{k} x^{n}\left(1-k^{-1} x\right)^{k} d x=n !\).

Short answer

The limit of the integral as \(k \to \infty\) is indeed \(n!\).

Step-by-step solution

Understand the Limit and Integral

The exercise requires us to prove that the integral of the function \(x^n (1-k^{-1}x)^k\) from \(0\) to \(k\) approaches \(n!\) as \(k\) approaches infinity. The expression \((1 - k^{-1} x)^k\) is reminiscent of the exponential function, suggesting this exercise involves properties of limits and definite integrals.

Change of Variables

Introduce a substitution to simplify the problem. Let \(t = k^{-1} x\), which implies \(x = kt\) and \(dx = k dt\). The integral limits change accordingly: when \(x = 0, t = 0\) and when \(x = k, t = 1\). Thus, the integral becomes:\[ \int_{0}^{1} (kt)^n (1-t)^k k \, dt = k^{n+1} \int_{0}^{1} t^n (1-t)^k \, dt.\]

Evaluate the New Integral

The integral \(\int_{0}^{1} t^n (1-t)^k \, dt\) is a Beta function, \(B(n+1, k+1)\), which is related to the Gamma function by:\[ B(n+1, k+1) = \frac{\Gamma(n+1) \Gamma(k+1)}{\Gamma(n+k+2)}\]

Analyze the Limit Using the Gamma Function

Since \(\Gamma(n+1) = n!\) and the approximation \(\Gamma(k+1) \approx \sqrt{2\pi k} \left(\frac{k}{e}\right)^k\) as \(k\) becomes large, apply Stirling's approximation. This simplifies the Beta function:\[ k^{n+1} B(n+1, k+1) \approx k^{n+1} \cdot \frac{n! \sqrt{2\pi k} \left(\frac{k}{e}\right)^k}{\Gamma(n+k+2)}\]As \(k \to \infty\), the integral approaches \(n!\).

Complete the Limit Calculation

Simplify the expression based on the Gamma and Beta function properties, and use the known limits:\[ k^{n+1} \frac{n!}{(n+k+1)(n+k)\cdots (k+1) k} \to n!\] as \(k \to \infty\). Therefore, you can conclude that the initial limit statement is true.\( \int_{0}^{1} t^n (1-t)^k \, dt \approx \frac{n!}{k^{n+1}}\)

Execute Final Verification

Verify your simplification leads to the original claim that:\[\lim _{k \rightarrow \infty} \int_{0}^{k} x^{n} \left(1-k^{-1} x\right)^{k} d x = n !\] using both the change of variables and analysis with Stirling's Approximation to confirm the integral approaches \(n!\).

Key concepts

Stirling's Approximation

Stirling's Approximation is a powerful mathematical technique used to estimate the factorial of a large number. The factorial function, which is denoted as \(n!\), grows very rapidly as \(n\) increases, making it difficult to calculate directly in certain contexts. Stirling's approximation provides a simpler formula to approximate \(n!\).

For a large \(n\), Stirling's approximation is given by:

• \( n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \)

Here, \(e\) is the base of the natural logarithm, and this approximation becomes more accurate as \(n\) increases.

This formula is particularly useful in complex analyses, such as asymptotic analysis, probability, and statistics, where understanding the behavior of factorials of large numbers is crucial. In the context of the original problem, Stirling's approximation is used to simplify parts of the expression involving the Gamma function as \(k\) approaches infinity.

Gamma Function

The Gamma function is an extension of the factorial function to complex numbers. It is denoted by \(\Gamma(n)\) and is defined for all complex numbers except the non-positive integers. For positive integers, the Gamma function relates to the factorial by:

• \(\Gamma(n) = (n-1)!\)

For any real number \(z\), the Gamma function can be represented by an integral:

• \(\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} \, dt\)

In the given problem, the integral transformation led to an equation resembling a Beta function which in turn was expressed in terms of Gamma functions. The relationship \(B(n+1, k+1) = \frac{\Gamma(n+1) \Gamma(k+1)}{\Gamma(n+k+2)}\) played a critical role. It helps simplify the analysis by leveraging properties of these functions, thereby reducing complex integrals to well-known mathematical terms.

Understanding the Gamma function's properties allows one to generalize uses beyond simple factorials, like simplifying limits involving factorials through transformations and integration.

Beta Function

The Beta function, denoted as \(B(x, y)\), is a special function that arises frequently in statistical distributions and calculus. It is closely related to the Gamma function through the identity:

• \( B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)} \)

The Beta function is often expressed as an integral:

• \( B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt \)

In the context of the discussed problem, after a change of variables, the integral naturally formed a Beta function. By translating this integral into Beta's terms, a complex problem can be approached through defined relationships. The transformation of the original integral to a Beta function effectively illustrates how numerous challenging integrals can sometimes be simplified using well-known functions and identities.

Employing the Beta function is a strategic move, especially when the goal is to express difficult integral calculations as simpler products and quotients of Gamma functions, which are easier to manipulate and approximate, particularly with functions like Stirling's approximation.