Asymptotics of alternating harmonic series with attenuation

We find the asymptotics of the series $\sum_{n=1}^\infty (-1)^n n^{-1} \exp(-t/n)$ as $t\to+\infty$. The answer is an oscillating function of $t$ dominated by $\exp(-(2πt)^{1/2})$. The intermediate step is to find the asymptotics of the two-dimensional Fourier transform $\hat F(ξ)$ of the function $F(x)=(1+\exp(\|x\|^2))^{-1}$ as $\|ξ\|\to\infty$.

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