# A Note on the 2F1 Hypergeometric Function - Mathematics > Classical Analysis and ODEs

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Abstract: The special case of the hypergeometric function $ {2}F {1}$ represents thebinomial series $1+x^{\alpha}=\sum {n=0}^{\infty}\:\alpha n\:x^{n}$ thatalways 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 everyconcrete case. In this note, using new approach, we reprove the convergence ofthe hypergeometric series $ {2}F {1}\alpha,\beta;\beta;x$ for $|x|<1$ andobtain new result on its convergence at point $x=-1$ for every integer$\alpha eq 0$. The proof is within a new theoretical setting based on the newmethod for reorganizing the integers and on the regular method for summation ofdivergent series.

Author: ** Armen Bagdasaryan**

Source: https://arxiv.org/