This final project discusses a decreasing sequence (an) of nonnegative terms, the- re exist subsequences (nk) of (n) such thatP1k=1(nk+1−nk)ank andP1n=1 an ha- ve the same convergence. Then for P1n=1 an converges if and only if P1k=1 qkaqk converges for 1 < q ∈ N. Using these results, an extension of the ratio test is formed. Keywords: Sequence, series, convergency, divergency