Correct Answer – Option 4 : None of these

Concept:

Cauchy’s Root test-

A series of positive terms ∑unis

(i) convergent if \(\mathop {\lim }\limits_{n \to \infty } {\left( {{u_n}} \right)^{\frac{1}{n}}} < 1\)

(ii) Divergent if \(\mathop {\lim }\limits_{n \to \infty } {\left( {{u_n}} \right)^{\frac{1}{n}}} > 1\)

(iii) Test fail if \(\mathop {\lim }\limits_{n \to \infty } {\left( {{u_n}} \right)^{\frac{1}{n}}} = 1\)

Observation

A necessary condition for convergence of an infinite series \(\sum {{u_n}} \) is that \(\mathop {\lim }\limits_{n \to \infty } {u_n}< 1\)

None of the options are correct. Hence, option 4 is the correct answer.