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.

  • $\begingroup$ Are you assuming the sequence to be $\ell^1$ here? $\endgroup$ – DanZimm Feb 8 '15 at 21:59
  • $\begingroup$ It is not necessarily $l_1$ $\endgroup$ – user168309 Feb 10 '15 at 1:40

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.


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