The series  of positive terms is convergent or divergent as  or < 1 then this test is known as

Concept:

D' Alembert's test: If  Σu_n is a positive term series, such that,  , then the series

(i) converges, if l < 1,

(ii) diverges, if l > 1,

(iii) the test fails, if l = 1

Additional Information

Comparison Test- If  and  are two positive term series, and k ≠ 0, a fixed positive real number and there exists a positive integer m such that u_n ≤ kv_n, 

∀ n ≥ m, then-

(i)  is convergent, if  is convergent and

(ii)  is divergent, if    is divergent.

Raabe's Test-

If  is a positive term series, such that  then the series,

(i) converges, if l > 1,

(ii) diverges, if l < 1,

(iii) the test fails, if l = 1.