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msc:mcqs_short_questions:real_analysis [2023/04/01 15:58] – Administrator | msc:mcqs_short_questions:real_analysis [2023/04/03 04:06] (current) – [Series of Numbers] Administrator | ||
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* (C) Additive identity | * (C) Additive identity | ||
* (D) Additive inverse of zero \\ <btn type=" | * (D) Additive inverse of zero \\ <btn type=" | ||
- | </ | + | </ |
+ | < | ||
2. Which one of them is not interval. | 2. Which one of them is not interval. | ||
* (A) $(1,2)$ | * (A) $(1,2)$ | ||
* (B) $\left(\frac{1}{2}, | * (B) $\left(\frac{1}{2}, | ||
* (C) $[3. \pi]$ | * (C) $[3. \pi]$ | ||
- | * (D) $(2\pi, | + | * (D) $(2\pi, |
- | </ | + | </ |
+ | < | ||
3. A number which is neither even nor odd is | 3. A number which is neither even nor odd is | ||
* (A) 0 | * (A) 0 | ||
* (B) 2 | * (B) 2 | ||
* (C) $2n$ such that $n \in \mathbb{Z}$ | * (C) $2n$ such that $n \in \mathbb{Z}$ | ||
- | * (D) $2\pi$ \\ <btn type=" | + | * (D) $2\pi$ \\ <btn type=" |
- | </ | + | </ |
+ | < | ||
4. A number which is neither positive nor negative is | 4. A number which is neither positive nor negative is | ||
* (A) 0 | * (A) 0 | ||
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* (C) 2 | * (C) 2 | ||
* (D) does not exist \\ <btn type=" | * (D) does not exist \\ <btn type=" | ||
- | </panel>< | + | </ |
- | ==== Sequence | + | ==== Sequence of Numbers ==== |
+ | < | ||
1. A convergent sequence has only ................ limit(s). | 1. A convergent sequence has only ................ limit(s). | ||
* (A) one | * (A) one | ||
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* (C) divergent. | * (C) divergent. | ||
* (D) bounded. \\ <btn type=" | * (D) bounded. \\ <btn type=" | ||
- | </ | + | </ |
- | 5. A sequence $\{\dfrac{1}{n} \}$ is | + | < |
+ | 5. A sequence $\left\{\dfrac{1}{n} | ||
* (A) bounded. | * (A) bounded. | ||
* (B) unbounded. | * (B) unbounded. | ||
* (C) divergent. | * (C) divergent. | ||
- | * (D) None of these. \\ <btn type=" | + | * (D) None of these. \\ <btn type=" |
- | </ | + | </ |
+ | < | ||
6. A sequence $\{s_n\}$ is said be Cauchy if for $\epsilon> | 6. A sequence $\{s_n\}$ is said be Cauchy if for $\epsilon> | ||
* (A) $|s_n-s_m|< | * (A) $|s_n-s_m|< | ||
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* (C) is convergent. | * (C) is convergent. | ||
* (D) is divergent. \\ <btn type=" | * (D) is divergent. \\ <btn type=" | ||
- | - A series $\sum a_n$ is convergent if and only if ..................... is convergent | + | </ |
- | * (A) $\{\sum_{k=1}^{\infty}a_k \}$ | + | ==== Series of Numbers ==== |
- | * (B) $\{\sum_{k=1}^{n}a_k \}$ | + | < |
- | * (C) $\{\sum_{n=1}^{\infty}a_k \}$ | + | 1. |
- | * (D) $\{ a_n \}$ | + | * (A) $s_n=\sum_{n=1}^\infty |
- | | + | * (B) $s_n=\sum_{k=1}^n a_k$ is convergent. |
- | * (A) it is bounded. | + | * (C) $s_n=\sum_{k=1}^n a_n$ is convergent. |
- | * (B) it may bounded. | + | * (D) $s_n=\sum_{k=1}^n a_k$ is divergent. \\ <btn type=" |
- | * (C) it is unbounded. | + | </ |
- | * (D) it may unbounded. | + | 2. If $\sum_{n=1}^\infty a_n$ converges then ........................... |
- | - If $\lim_{n\to\infty} a_n=0$, then $\sum a_n$ ................ | + | * (A) $\lim_{n\to \infty} a_n=0$. |
+ | * (B) $\lim_{n\to \infty} a_n=1$. | ||
+ | * (C) $\lim_{n\to \infty} a_n \neq 0$ | ||
+ | * (D) $\lim_{n\to \infty} a_n$ exists. \\ <btn type=" | ||
+ | </ | ||
+ | < | ||
+ | 3. If $\lim_{n\to \infty} a_n \neq 0$, then $\sum_{n=1}^\infty | ||
+ | * (A) is convergent. | ||
+ | * (B) may convergent. | ||
+ | * (C) is divergent | ||
+ | * (D) is bounded. \\ <btn type=" | ||
+ | </ | ||
+ | < | ||
+ | 4. A series $\sum_{n=1}^\infty \left( 1+\frac{1}{n} \right)$ is .................... | ||
+ | * (A) convergent. | ||
+ | * (B) divergent. | ||
+ | * (C) constant. | ||
+ | * (D) none of these \\ <btn type=" | ||
+ | </ | ||
+ | 5. Let $\sum a_n$ be a series of non-negative terms. Then it is convergent if its sequence of partial sum ............... | ||
+ | * (A) is bounded. | ||
+ | * (B) may bounded. | ||
+ | * (C) is unbounded. | ||
+ | * (D) is divergent. \\ <btn type=" | ||
+ | </ | ||
+ | | ||
* (A) is convergent. | * (A) is convergent. | ||
* (B) is divergent. | * (B) is divergent. | ||
* (C) may or may not convergent | * (C) may or may not convergent | ||
- | * (D) none of these | + | * (D) none of these \\ <btn type=" |
- | - A series $\sum \frac{1}{n^p}$ is convergent if | + | </ |
- | * (A) $p\geq 1$. | + | 7. A series $\sum \frac{1}{n^p}$ is convergent if |
- | * (B) $p\leq 1$. | + | * (A) $p\leq 1$. |
- | * (C) $p>1$. | + | * (B) $p\geq 1$. |
- | * (D) $p<1$. | + | * (C) $p<1$. |
- | - If a sequence $\{a_n\}$ is convergent then the series $\sum a_n$ ................ | + | * (D) $p>1$. \\ <btn type=" |
+ | </ | ||
+ | 8- If a sequence $\{a_n\}$ is convergent then the series $\sum a_n$ ................ | ||
* (A) is convergent. | * (A) is convergent. | ||
* (B) is divergent. | * (B) is divergent. | ||
* (C) may or may not convergent | * (C) may or may not convergent | ||
- | * (D) none of these | + | * (D) none of these \\ <btn type=" |
- | | + | </ |
+ | | ||
* (A) $\{a_n\}$ is convergent. | * (A) $\{a_n\}$ is convergent. | ||
* (B) $\{a_n\}$ is decreasing. | * (B) $\{a_n\}$ is decreasing. | ||
* (C) $\{a_n\}$ is bounded. | * (C) $\{a_n\}$ is bounded. | ||
- | * (D) $\{a_n\}$ is decreasing and $\lim a_n=0$. | + | * (D) $\{a_n\}$ is decreasing and $\lim a_n=0$. |
- | | + | </ |
+ | 10. An series $\sum a_n$ is said to be absolutely convergent if | ||
* (A) $\left| \sum a_n \right|$ is convergent. | * (A) $\left| \sum a_n \right|$ is convergent. | ||
* (B) $\left| \sum a_n \right|$ is convergent but $\sum a_n$ is divergent. | * (B) $\left| \sum a_n \right|$ is convergent but $\sum a_n$ is divergent. | ||
* (C) $\sum |a_n|$ is convergent. | * (C) $\sum |a_n|$ is convergent. | ||
- | * (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent. | + | * (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent. |
- | | + | </ |
- | | + | 11. A series $\sum a_n$ is convergent if and only if ..................... is convergent |
- | | + | * (A) $\{\sum_{k=1}^{\infty}a_k \}$ |
- | | + | * (B) $\{\sum_{k=1}^{n}a_k \}$ |
- | | + | * (C) $\{\sum_{n=1}^{\infty}a_k \}$ |
- | - If $\lim_{x \to c}f(x)=L$, then ............ sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$. | + | * (D) $\{ a_n \}$ \\ <btn type=" |
- | | + | </ |
- | | + | ==== Limit of functions ==== |
- | | + | < |
- | | + | 1. A number $L$ is called limit of the function $f$ when $x$ approaches to $c$ if for all $\varepsilon> |
- | | + | |
- | | + | |
- | | + | |
- | | + | |
- | | + | </ |
- | | + | 2. If $\lim_{x \to c}f(x)=L$, then .............. sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$. |
- | | + | |
- | | + | |
- | | + | |
- | | + | |
- | | + | </ |
- | | + | 3. Let $f(x)=\frac{x^2-5x+6}{x-3}$, |
- | | + | |
- | | + | |
- | | + | |
+ | | ||
+ | </ | ||
+ | ==== Riemann Integrals ==== | ||
+ | < | ||
+ | 1. Which one is not partition of interval $[1,5]$. | ||
+ | * (A) $\{1,2,3,5 \}$ | ||
+ | * (B) $\{1, | ||
+ | * (C)$\{1,1.1,5 \}$ | ||
+ | * (D) $\{1,2.1,3,4,5.5 \}$ \\ <btn type=" | ||
+ | </ | ||
+ | 2. What is norm of partition $\{0, | ||
+ | * (A) $10$ | ||
+ | * (B) $3$ | ||
+ | * (C) $3.8$ | ||
+ | * (D) $0.1$ \\ <btn type=" | ||
+ | </ | ||