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A Simple Proof that Γ(n+1) = n!
sos440
2012. 3. 12. 16:26
이런 거라도 올려야 블로그가 정체되지 않겠지요? 으헣헣 ;ㅅ;
Tonelli's theorem enables us to exchange the order of integration and summation of a sequence of nonnegative functions. Thus for \( 0 < r < 1 \),
\begin{align*} \sum_{n=0}^{\infty} \frac{r^n}{n!} \int_{0}^{\infty} x^n e^{-x} \; dx & = \int_{0}^{\infty} \sum_{n=0}^{\infty} \frac{(rx)^n}{n!} e^{-x} \; dx = \int_{0}^{\infty} e^{-(1 - r)x} \; dx \\ & = \frac{1}{1 - r} = \sum_{n=0}^{\infty} r^{n}. \end{align*}
Tonelli's theorem enables us to exchange the order of integration and summation of a sequence of nonnegative functions. Thus for \( 0 < r < 1 \),
\begin{align*} \sum_{n=0}^{\infty} \frac{r^n}{n!} \int_{0}^{\infty} x^n e^{-x} \; dx & = \int_{0}^{\infty} \sum_{n=0}^{\infty} \frac{(rx)^n}{n!} e^{-x} \; dx = \int_{0}^{\infty} e^{-(1 - r)x} \; dx \\ & = \frac{1}{1 - r} = \sum_{n=0}^{\infty} r^{n}. \end{align*}