A solution of one problem of complex integration

Abstract

In this paper the following identity

$$(\sqrt\pi+\int_0^ie^{-1/t^2}dt)e^{-1}=i\Big(1-{2^1\over 1}+{2^2\over 1\cdot 3}-{2^3\over 1\cdot 3\cdot 5}\ldots\Big)$$

is proved, where the integration is done over a curve with tangent vector at 0 toward the positive part of $x$-axis.