Indefinite Integration of the Gamma Integral
and Related Statistical Applications
2.4 Answer to the Unsolved Puzzle
What is wrong with treating $c$ as infinity? Negligence of the relative power of $c$ and $n$ to asymptotic infinity drives an incorrect conclusion in (2.17), that all of the weighting factor ${{w}_{i}}$ is one, and thus $\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{{{w}_{i}}{{\beta }_{i}}}$ approaches infinity. As (2.14) indicates, for any given value of $c$, we can always find a larger number $N$ that makes ${w}_{N+j}$, where $j\in {{\mathbb{Z}}^{+}_{0}}$, approach 0. By taking a derivative, we know that the integrand function ${{x}^{N}}\exp(-x)$ has the maximum value at $x=N$. Thus, the integral value at $\left[ 0,c \right]$ as a ratio of the same integral value at the overall domain $\left[ 0,\infty \right]$ will approach 0 if $c\ll N$. This means, ${w}_{i}$ cannot always be one because $N$ will eventually overpower $c$ in creating the weighting factor.
This conclusion explains why the infinite series based on the Taylor series (1.2) or Kummer's confluent hypergeometric function (1.4) was not widely recognized as a solution to the gamma integral. Employing the "$h$" factorization method, we can actually establish the identity between "$h$" and the gamma function, with an understanding that the infinite upper limit $c$ should be regarded as a finite number. Once we realize the finite property of $c$, we are on firm ground to understand the "$h$" factorization method presented in (2.2) as a general solution to the gamma integral.
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