text: # If $$\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$$, then the series $$\sum {{a_n}}$$ is:
  
  This question was previously asked in
  UP TGT Mathematics 2016 Official Paper
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  1. Oscillatory
  2. Divergent
  3. Convergent
  4. None of these
  
  Option 3 : Convergent
  Free
  CT 1: हिन्दी (आदिकाल)
  6912
  10 Questions 40 Marks 10 Mins
  
  ## Detailed Solution
  
  Concept:
  
  Consider the infinite series ∑un = u1 + u2 + u3 + … ∞
  
  And let the sum of the first n terms be Sn = u1 + u2 + u3 + … + un;
  
  If Sn tends to a zero or finite limit as n → ∞, the series is said to be convergent
  
  If Sn tends to ± ∞ as n → ∞, the series is said to be divergent
  
  If Sn does not tend to a unique limit as n → ∞, the series is said to be oscillatory or non-convergent
  
  Calculation:
  
  Given:
  
  $$\mathop {\lim }\limits_{n \to \infty } {a_n} = 0$$
  
  Then series $$\sum {{a_n}}$$ is tending to zero
  
  Hence, the given sequence is Covergent.
  
  Important Points
  
  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 $$\mathop {\lim }\limits_{n \to \infty } {\left( {{u_n}} \right)^{\frac{1}{n}}} = 1$$