Behaviour of sum of $2^\kappa$ for all $\kappa<\lambda$ when $\lambda$ is singular

Question

What can we say about the conditions under which $\sum\limits_{\kappa<\lambda}2^\kappa \leq \lambda$ holds when $\lambda$ is singular?

Answer 1

Note that if $\lambda$ is singular, and $\kappa$ is its confinality then $2^\kappa\neq\lambda$. And in fact the same inequality holds for every cardinal above $\kappa$ for the same reasons. König's theorem.

So there is really just one way for this inequality to hold. $\lambda$ has to be a strong limit cardinal.