Can someone help me the summation of the given series.
Negative of the above summation looks similar to the expansion of the $\log(1-x)$ without the floor.
This is OEIS A$006218$. No closed form is known; the problem of finding a precise asymptotic estimate is the Dirichlet divisor problem. You’ll find references, asymptotic estimates, and other information at the two links and here.
A useful tip: I simply calculated the first half-dozen terms and ran them through The On-Line Encyclopedia of Integer Sequences (OEIS); that’s worth a try any time you want information about an integer sequence.