Library / Functions of a Real Variable / Chapter V / § 4
Example
Tag 08GA, permanent. Cite it as [Bourbaki, Tag 08GA].
Functions of a Real Variable, chapter V, § 4. APPLICATION TO SERIES WITH POSITIVE TERMS
Consider the hypergeometric series , with general term
u n = α ( α + 1 ) … ( α + n − 1 ) β ( β + 1 ) … ( β + n − 1 ) 1.2 … n . γ ( γ + 1 ) … ( γ + n − 1 )
u_n = \frac{\alpha(\alpha+1)\ldots(\alpha+n-1)\,\beta(\beta+1)\ldots(\beta+n-1)}{1.2\ldots n.\gamma(\gamma+1)\ldots(\gamma+n-1)}
u n = 1.2 … n . γ ( γ + 1 ) … ( γ + n − 1 ) α ( α + 1 ) … ( α + n − 1 ) β ( β + 1 ) … ( β + n − 1 ) where α , β , γ \alpha, \beta, \gamma α , β , γ ≤ 0 \leq 0 ≤ 0 u n u_n u n > 0 > 0 > 0 < 0 < 0 < 0
u n + 1 u n = ( α + n ) ( β + n ) ( n + 1 ) ( γ + n ) = ( 1 + α + β n + α β n 2 ) ( 1 + γ + 1 n + γ n 2 ) − 1 = 1 + α + β − γ − 1 n + α β − ( α + β ) ( γ + 1 ) + γ 2 + γ + 1 n 2 + o ( 1 n 2 ) .
\begin{align*}
\frac{u_{n+1}}{u_n} &= \frac{(\alpha+n)(\beta+n)}{(n+1)(\gamma+n)} \\
&= \left(1 + \frac{\alpha+\beta}{n} + \frac{\alpha\beta}{n^2}\right) \left(1 + \frac{\gamma+1}{n} + \frac{\gamma}{n^2}\right)^{-1} \\
&= 1 + \frac{\alpha+\beta-\gamma-1}{n} \\
&\quad + \frac{\alpha\beta-(\alpha+\beta)(\gamma+1)+\gamma^2+\gamma+1}{n^2} + o\left(\frac{1}{n^2}\right).
\end{align*}
u n u n + 1 = ( n + 1 ) ( γ + n ) ( α + n ) ( β + n ) = ( 1 + n α + β + n 2 α β ) ( 1 + n γ + 1 + n 2 γ ) − 1 = 1 + n α + β − γ − 1 + n 2 α β − ( α + β ) ( γ + 1 ) + γ 2 + γ + 1 + o ( n 2 1 ) . Raabe's test shows that the series converges for α + β < γ \alpha + \beta < \gamma α + β < γ α + β > γ \alpha + \beta > \gamma α + β > γ α + β = γ \alpha + \beta = \gamma α + β = γ
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Transcribed from the printed volume. Not translated.