Consider a positive square summable sequence
$$\sum_{k=1}^{K} a_k^2 < +\infty,$$
where $K$ can be infinity. Can we have any estimate or upper bound of the $l_1$ summation of the sequence?($\sum_{k=1}^{K} a_k$)
Of course one answer is to use the Cauchy-Schwarz inequality, but that is not a tight bound.
Best Answer
We can take $a_n=\frac{1}{n}$ so we have $\sum a_n^2 <\infty$ and $\sum a_n =\infty$. So I think you wont find some nice bound for infinity sum.