Sequences with tails in $l^1$

Suppose that $a_n \geq a_{n+1} > 0$ and $\sum_{n \geq 0}a_n < \infty$. If the tails of the sequence

$t_N = \sum_{n \geq N}a_n$

are also in $l^1$, can anything be said about the rate at which $a_n \rightarrow 0$?

Note that if $a_n = \frac{1}{(1+\varepsilon)^n}$, then $t_n = \frac{1}{\varepsilon\cdot (1+\varepsilon)^n}$ so $a_n$ has summable tails.