Proof. [024E]

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Proof.

We compute

\displaystyle\int_{1}^{x}(1-y^{-2})^{-1/n}dy=\int_{1}^{x}\sum_{k=0}^{\infty}\frac{(\frac{1}{n})_{k}}{k!}y^{-2k}dy=y\sum_{k=0}^{\infty}\frac{(\frac{1}{n})_{k}}{k!}\frac{y^{-2k}}{(-2k+1)}\bigg|_{y=1}^{y=x}.

The Lemma follows from the observation $\frac{(-\frac{1}{2})_{k}}{(\frac{1}{2})_{k}}=-\frac{1}{2k-1}.$ ∎