The nth term seems to be √(½.2ⁿ) although this is not the series as written in full in the question, which contains inconsistencies. If we start with n=0, the first few terms would be:
√½, √(½.2), √(½.4), √(½.8),...
The given series in the question could not be represented in sigma notation because of the inconsistencies:
there is no √(½.2) term, and the last term √⅛ cannot be written as √½.2ⁿ for n≥0, because if ⅛=½.2ⁿ means that 2ⁿ=¼, so n=-2.
If we assume that the nth term is √(½.2ⁿ) this can be written √2ⁿ⁻¹, then the sum of the series is ∑√2ⁿ⁻¹ or ∑2^(½(n-1)).