# 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
View all UP TGT Papers >
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 $$\mathop {\lim }\limits_{n \to \infty } {\left( {{u_n}} \right)^{\frac{1}{n}}} = 1$$
The text passage contains a mathematical question about the convergence of a series, followed by multiple-choice options and a detailed solution that explains the correct answer.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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
View all UP TGT Papers >
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 $$\mathop {\lim }\limits_{n \to \infty } {\left( {{u_n}} \right)^{\frac{1}{n}}} = 1$$
user
The text passage contains a mathematical question about the convergence of a series, followed by multiple-choice options and a detailed solution that explains the correct answer.

TYPE: SINGLE_QUESTION_WITH_ANSWER