How fast does a sequence with finite sum go to zero? Suppose $z_n$ is a nonnegative sequence, monotonically decreasing to zero, and $$\sum_{i=1}^{\infty} z_i < 1.$$ Is it possible to translate this into a bound on the how small $z_i$ is? For example, it seems natural to guess that $z_i < z_0 \alpha^i$ for some $\alpha$ that depends on $z_0$. Is this true? If not, can anything about the decay of the sequence be inferred from the sum bound?

At the very least, such an $\alpha$ might not exist.  As an example, consider $z_i = 1/i^{1+\epsilon}$ for any (small) $\epsilon > 0$.
Such an $\alpha<1$ exists exactly when $\sum z_i$ can be shown to converge by the ratio-test.
