Abstract

The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty}\left(\:\begin{matrix}\alpha\\n\end{matrix}\:\right)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$, depends on the values of $\alpha$ and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series for $|x|<1$ and obtain new result on its convergence at point $x=-1$ for every integer $\alpha\neq 0$, that is we prove it for the function $_{2}F_{1}(\alpha,\beta;\beta;x)$. The proof is within a new theoretical setting based on a new method for reorganizing the integers and on the original regular method for summation of divergent series.

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