The series \(\sum {{u_n}} \) of positive terms is convergent or divergent as \(\mathop {\lim }\limits_{n \to \infty } \frac{{{u_n}}}{{{u_{n + 1}}}} > 1\) or < 1 then this test is known as

Concept:

D' Alembert's test: If  Σu_n is a positive term series, such that,  \(\mathop {\lim }\limits_{n \to \infty } \frac{{{u_n}}}{{{u_{n + 1}}}} = l\), 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 \(\sum u_n\) and \(\sum v_n\) 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) \(\sum u_n\) is convergent, if \(\sum v_n\) is convergent and

(ii) \(\sum v_n\) is divergent, if  \(\sum u_n\)  is divergent.

Raabe's Test-

If \(\sum u_n\) is a positive term series, such that \(\mathop {\lim }\limits_{n \to \infty }n (\frac{{{u_n}}}{{{u_{n + 1}}}}-1) = 1\) then the series,

(i) converges, if l > 1,

(ii) diverges, if l < 1,

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