Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
Last revisionBoth sides next revision
msc:mcqs_short_questions:real_analysis [2023/04/01 15:58] – Administratormsc:mcqs_short_questions:real_analysis [2023/04/03 04:03] – [Sequence of Numbers] Administrator
Line 30: Line 30:
     * (C) Additive identity     * (C) Additive identity
     * (D) Additive inverse of zero \\ <btn type="link" collapse="a1">See Answer</btn><collapse id="a1" collapsed="true">(B): zero is neither positive not negative</collapse>     * (D) Additive inverse of zero \\ <btn type="link" collapse="a1">See Answer</btn><collapse id="a1" collapsed="true">(B): zero is neither positive not negative</collapse>
-</panel><panel>+</panel> 
 +<panel>
  2. Which one of them is not interval.  2. Which one of them is not interval.
     * (A) $(1,2)$     * (A) $(1,2)$
     * (B) $\left(\frac{1}{2},\frac{1}{3} \right)$     * (B) $\left(\frac{1}{2},\frac{1}{3} \right)$
     * (C) $[3. \pi]$     * (C) $[3. \pi]$
-    * (D) $(2\pi,180)$ \\ <btn type="link" collapse="a2">See Answer</btn><collapse id="a2" collapsed="true">(B): In interval (a,b), a<b.</collapse> +    * (D) $(2\pi,180)$ \\ <btn type="link" collapse="a2">See Answer</btn><collapse id="a2" collapsed="true">(B): In interval $(a,b)$, $a<b$ but $\frac{1}{2}>\frac{1}{3}$.</collapse> 
- </panel><panel>+ </panel> 
 +<panel>
  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="link" collapse="a3">See Answer</btn><collapse id="a3" collapsed="true">(D): $2\pi$ is not an integer.</collapse> +    * (D) $2\pi$ \\ <btn type="link" collapse="a3">See Answer</btn><collapse id="a3" collapsed="true">(D): Integers can only be even or odd but $2\pi$ is not an integer.</collapse> 
- </panel><panel>+ </panel> 
 +<panel>
  4. A number which is neither positive nor negative is   4. A number which is neither positive nor negative is 
     * (A) 0     * (A) 0
Line 78: Line 81:
     * (C) 2     * (C) 2
     * (D) does not exist \\ <btn type="link" collapse="a9">See Answer</btn><collapse id="a9" collapsed="true">(C): In tabular form $A=\{1, 2 \}$ and set of upper bouds is $\{2,3,4,... \}$. Now supremum is least upper bound $2$. </collapse>     * (D) does not exist \\ <btn type="link" collapse="a9">See Answer</btn><collapse id="a9" collapsed="true">(C): In tabular form $A=\{1, 2 \}$ and set of upper bouds is $\{2,3,4,... \}$. Now supremum is least upper bound $2$. </collapse>
- </panel><panel> + </panel> 
- ==== Sequence and Series of Numbers ====+==== Sequence of Numbers ==== 
 + <panel>
  1. A convergent sequence has only ................ limit(s).  1. A convergent sequence has only ................ limit(s).
     * (A) one     * (A) one
Line 103: Line 107:
     * (C) divergent.     * (C) divergent.
     * (D) bounded. \\ <btn type="link" collapse="a13">See Answer</btn><collapse id="a13" collapsed="true">(D): As $|(-1)^n| = 1 < 1.1$ for all $n \in \mathbb{N}, therefore it is bounded.$</collapse>     * (D) bounded. \\ <btn type="link" collapse="a13">See Answer</btn><collapse id="a13" collapsed="true">(D): As $|(-1)^n| = 1 < 1.1$ for all $n \in \mathbb{N}, therefore it is bounded.$</collapse>
- </panel><panel> + </panel> 
-5. A sequence $\{\dfrac{1}{n} \}$ is+<panel> 
 +5. A sequence $\left\{\dfrac{1}{n} \right\}$ is
     * (A) bounded.     * (A) bounded.
     * (B) unbounded.     * (B) unbounded.
     * (C) divergent.     * (C) divergent.
-    * (D) None of these. \\ <btn type="link" collapse="a14">See Answer</btn><collapse id="a14" collapsed="true">(A): As $\{\dfrac{1}{n} \}$ is convergent, it is bounded or it is easy to see $|\dfrac{1}{n}| \leq 1$ for all $n \in \mathbb{N}.</collapse> +    * (D) None of these. \\ <btn type="link" collapse="a14">See Answer</btn><collapse id="a14" collapsed="true">(A): As $\left\{\dfrac{1}{n} \right\}$ is convergent, it is bounded or it is easy to see $\left|\dfrac{1}{n} \right| \leq 1$ for all $n \in \mathbb{N}$.</collapse> 
- </panel><panel>+ </panel> 
 +<panel>
 6. A sequence $\{s_n\}$ is said be Cauchy if for $\epsilon>0$, there exists positive integer $n_0$ such that  6. A sequence $\{s_n\}$ is said be Cauchy if for $\epsilon>0$, there exists positive integer $n_0$ such that 
     * (A) $|s_n-s_m|<\epsilon$ for all $n,m>0$.     * (A) $|s_n-s_m|<\epsilon$ for all $n,m>0$.
Line 163: Line 169:
     * (C) is convergent.     * (C) is convergent.
     * (D) is divergent. \\ <btn type="link" collapse="214">See Answer</btn><collapse id="214" collapsed="true">(D): Every subsequence of convergent sequence converges to the same limit.</collapse>     * (D) is divergent. \\ <btn type="link" collapse="214">See Answer</btn><collapse id="214" collapsed="true">(D): Every subsequence of convergent sequence converges to the same limit.</collapse>
-  - A series $\sum a_n$ is convergent if and only if ..................... is convergent  + </panel> 
-    * (A) $\{\sum_{k=1}^{\infty}a_k \}$ +==== Series of Numbers ==== 
-    * (B) $\{\sum_{k=1}^{n}a_k \}$ +<panel> 
-    * (C) $\{\sum_{n=1}^{\infty}a_k \}$ +1.  A series $\sum_{n=1}^\infty a_n$ is said to be convergent if the sequence $\{ s_n \}$, where .................. 
-    * (D) $\{ a_n \}$ +    * (A) $s_n=\sum_{n=1}^\infty a_n$ is convergent. 
-  - Let $\sum a_n$ be a series of non-negative terms. Then it is convergent if ........... +    * (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="link" collapse="301">See Answer</btn><collapse id="301" collapsed="true">(B): Series is convergent if its sequence of partial sume is convergent.</collapse> 
-    * (C) it is unbounded. +</panel><panel> 
-    * (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="link" collapse="302">See Answer</btn><collapse id="302" collapsed="true">(A)</collapse> 
 +</panel> 
 +<panel> 
 +3.  If  $\lim_{n\to \infty} a_n \neq 0$, then $\sum_{n=1}^\infty a_n$    ........................... 
 +    *  (A) is convergent. 
 +    * (B) may convergent. 
 +    * (C) is divergent 
 +    * (D) is bounded. \\ <btn type="link" collapse="303">See Answer</btn><collapse id="303" collapsed="true">(C): It is called divergent test</collapse> 
 +</panel> 
 +<panel> 
 +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="link" collapse="304">See Answer</btn><collapse id="304" collapsed="true">(B): As $\lim_{n\to \infty}\,\left( 1+\frac{1}{n} \right)=1\ne 0$, therefore by divergent test, the given series is divergent.</collapse> 
 +</panel><panel> 
 +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="link" collapse="305">See Answer</btn><collapse id="305" collapsed="true">(A): If  $\sum a_n$ is a non-negative terms series, then its sequence of partial sum is increasing. A monotone sequence of partial sume is convergent, if it is bounded.</collapse> 
 + </panel><panel> 
 + 6. If $\lim_{n\to\infty} a_n=0$, then $\sum a_n$ ................
     * (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="link" collapse="306">See Answer</btn><collapse id="306" collapsed="true">(C): If  $\sum a_n$ is convergent, then $\lim_{n\to\infty} a_n=0$ but converse may not true. e.g., $\sum \frac{1}{n}$ is divergent. </collapse> 
-  - A series $\sum \frac{1}{n^p}$ is convergent if + </panel><panel> 
-    * (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="link" collapse="307">See Answer</btn><collapse id="307" collapsed="true">(D): The p-series test, it can be proved easily by Cauchy condensation test.  </collapse> 
 + </panel><panel> 
 + 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="link" collapse="308">See Answer</btn><collapse id="308" collapsed="true">(C): The p-series test, it can be proved easily by Cauchy condensation test. </collapse> 
-  - An alternating series $\sum (-1)^n a_n$, where $a_n\geq 0$ for all $n$, is convergent if+ </panel><panel> 
 + 9. An alternating series $\sum (-1)^n a_n$, where $a_n\geq 0$ for all $n$, is convergent if
     * (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 $\lim a_n=0$. \\ <btn type="link" collapse="309">See Answer</btn><collapse id="309" collapsed="true">(B): Its called alternating series test. </collapse> 
-  - An series $\sum a_n$ is said to be absolutely convergent if+ </panel><panel> 
 +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. \\ <btn type="link" collapse="310">See Answer</btn><collapse id="310" collapsed="true">(C): It is definition of absolutely convergent. </collapse> 
-  - A number $L$ is called limit of the function $f$ when $x$ approaches to $c$ if for all $\varepsilon>0$, there exist $\delta>0$ such that ......... whenever $0<|x-c|<\delta$. + </panel> 
-    - $|f(x)-L| > \varepsilon$ +==== Limit of functions ==== 
-    - $|f(x)-L| < \varepsilon$ +<panel> 
-    - $|f(x)-L| \leq \varepsilon$ +1. A number $L$ is called limit of the function $f$ when $x$ approaches to $c$ if for all $\varepsilon>0$, there exist $\delta>0$ such that ......... whenever $0<|x-c|<\delta$. 
-    - $|f(x)-L| \geq \varepsilon$  +    * (A) $|f(x)-L| > \varepsilon$ 
-  - If $\lim_{x \to c}f(x)=L$, then ............ sequence $\{x_n\}$ such that $x_n \to c$, when $n\to \infty$, one has $\lim_{n \to \infty}f(x_n)=L$. +    * (B) $|f(x)-L| < \varepsilon$ 
-    - for some +    * (C) $|f(x)-L| \leq \varepsilon$ 
-    - for every +    * (D) $|f(x)-L| \geq \varepsilon$  \\ <btn type="link" collapse="401">See Answer</btn><collapse id="401" collapsed="true">(B): It is a definition of limit of functions. </collapse> 
-    - for few + </panel><panel> 
-    - none +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 $\lim_{n \to \infty}f(x_n)=L$. 
-  - Let $f(x)=\frac{x^2-5x+6}{x-3}$, then $\lim_{x\to 3}f(x)=$........... +    * (A) for some 
-    - $-1$ +    * (B) for every 
-    - $0$ +    * (C) for few 
-    - $1$ +    * (D) none of these \\ <btn type="link" collapse="402">See Answer</btn><collapse id="402" collapsed="true">(B) </collapse> 
-    - doesn't exist. + </panel><panel> 
-  - Which one is not partition of interval $[1,5]$. +3. Let $f(x)=\frac{x^2-5x+6}{x-3}$, then $\lim_{x\to 3}f(x)=$........... 
-    - $\{1,2,3,5 \}$ +    * (A) $-1$ 
-    - $\{1,3,3.5,5 \}$ +    * (B) $0$ 
-    - $\{1,1.1,5 \}$ +    * (C) $1$ 
-    - $\{1,2,3,4,5 \}$  +    * (D) doesn't exist. \\ <btn type="link" collapse="403">See Answer</btn><collapse id="403" collapsed="true">(C): $\lim_{x\to 3}f(x)=\frac{x^2-5x+6}{x-3}=\lim_{x\to 3}\frac{(x-2)(x-3)}{x-3}$ $=\lim_{x\to 3}(x-2) = 1$. </collapse> 
-  - What is norm of partition $\{0,3,3.1,3.2,7,10 \}$ of interval $[0,10]$. + </panel> 
-    - $10$ +==== Riemann Integrals ==== 
-    - $3$ +<panel> 
-    - $3.8$ +1. Which one is not partition of interval $[1,5]$. 
-    - $0.1$+   * (A) $\{1,2,3,5 \}$ 
 +   * (B) $\{1,3,3.5,5 \}$ 
 +   * (C)$\{1,1.1,5 \}$ 
 +   * (D) $\{1,2.1,3,4,5.5 \}$ \\ <btn type="link" collapse="601">See Answer</btn><collapse id="601" collapsed="true">(D): All points must be between $1$ and $5$.</collapse> 
 + </panel><panel> 
 +2. What is norm of partition $\{0,3,3.1,3.2,7,10 \}$ of interval $[0,10]$. 
 +   * (A) $10$ 
 +   * (B) $3$ 
 +   * (C) $3.8$ 
 +   * (D) $0.1$ \\ <btn type="link" collapse="602">See Answer</btn><collapse id="602" collapsed="true">(C): Maximum distance between any two points of the partition is norm, which is $7-3.2=3.8$.</collapse> 
 +</panel>