Infinite series question

Suppose, x_i in R, 0 < x_i < 1, sum_i x_i = oo, and

A_n = sum_{i <= i <= n} x_i, so that A_n -> oo.

B_n = sum_{1 <= i <= n} x_i exp[-A_n * xi].

Are there any known results under which conditions on (x_i), we have that B_n -> 0 as n -> oo ?

Thanks.
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