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SERIES OF REAL NUMBERS–MCQ–LEVEL–2

SERIES OF REAL NUMBERS–MCQ–LEVEL–2

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Question 1 of 10

1. Question
For the series $\displaystyle{\sum_{n = 1}^\infty {{P_n}} ,}$ where P_n = n!/nⁿ, the series
- Diverges
- Convergent
- Oscillates
- Monotonically decreasing
Correct

Incorrect
Question 2 of 10

2. Question
Let $b_n = 2^{{{( - 1)}^n}n}$ for $n \in \mathbb{N}.$ Then the series $\displaystyle{\sum_{n = 1}^\infty {{b_n}} }$ is
- Convergent
- Unbounded
- Divergent
- Oscillating
Correct

Incorrect
Question 3 of 10

3. Question
Let ${a_n} = \dfrac{1}{{n^2}}$ then the sequence of partial sums of the series $\displaystyle{\sum_{n = 1}^\infty {{a_n}}}$ is
- Cauchy
- Divergent
- Convergent to $\pi$
- Convergent to $\dfrac{\pi }{6}$
Correct

Incorrect
Question 4 of 10

4. Question
The series $\dfrac{{1000}}{1} + \dfrac{{1000 \cdot 1001}}{{1 \cdot 3}} + \dfrac{{1000 \cdot 1001 \cdot 1002}}{{1 \cdot 3 \cdot 5}} + ...$
- Converges
- Bounded by 101
- Unbounded
- Divergent
Correct

Incorrect
Question 5 of 10

5. Question
Let ${a_n} = {n^{\left( {n + \dfrac{1}{n}} \right)}},{b_n} = {\left( {n + \dfrac{1}{n}} \right)^n}$ then the series $\displaystyle{\sum_{n = 1}^\infty {\dfrac{{{a_n}}}{{{b_n}}}} }$
- converges to 1
- diverges
- converges to $e$
- converges to $e^{-1}$
Correct

Incorrect
Question 6 of 10

6. Question
Consider the series $\displaystyle{\sum_{n \ge 1} {\left( {{2^{1/2}} - {2^{1/3}}} \right)} \left( {{2^{1/2}} - {2^{1/5}}} \right)...\left( {{2^{1/2}} - {2^{1/(2n + 1)}}} \right),}$ then it
- oscillates between $2^{1/10}$ and $2^{1/20}$
- is cauchy
- is divergent to $+ \infty$
- is divergent to $- \infty$
Correct

Incorrect
Question 7 of 10

7. Question
Let ${a_n} = \dfrac{{{n^2}}}{{{{\left( {2 + \dfrac{1}{n}} \right)}^n}}},$ then the series $\displaystyle{\sum_{n = 1}^\infty {{a_n}} }$
- Diverges to $-\infty$
- Diverges to $+ \infty$
- Converges
- None of above holds
Correct

Incorrect
Question 8 of 10

8. Question
The series $\sqrt 2 + \sqrt {2 - \sqrt 2 } + \sqrt {2 - \sqrt {2 + \sqrt 2 } } + \sqrt {2 - \sqrt {2 + \sqrt {2 + \sqrt 2 } } } +...$ is
- Convergent
- Divergent
- If this series is $\displaystyle{\sum_{n = 1}^\infty {{a_n}} }$ then $\displaystyle\lim_{n \to \infty } {a_n} \ne 0$
- This series is unbounded above
Correct

Incorrect
Question 9 of 10

9. Question
Let ${a_n} = {{{{(n!)}^2}} \mathord{\left/ {\vphantom {{{{(n!)}^2}} {(2n)!}}} \right. \kern-\nulldelimiterspace} (2n)!}$ for $n \in \mathbb{N}.$ Then the series $\displaystyle{\sum_{n = 1}^\infty {{a_n}} }$ is
- Convergent
- Oscillating
- Divergent
- Unbounded
Correct

Incorrect
Question 10 of 10

10. Question
Let ${a_n} = \sqrt {n + 1} - \sqrt n.$ Then the series $\displaystyle{\sum_{n = 1}^\infty {{a_n}}}$ is
- Convergent to $\sqrt{2}$
- Convergent to $\sqrt{3}-\sqrt{2}$
- Convergent to $e$
- Divergent
Correct

Incorrect

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