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msc:mcqs_short_questions:real_analysis [2023/04/01 18:41] – [Riemann Integrals] Administratormsc:mcqs_short_questions:real_analysis [2023/04/03 04:03] – [Sequence of Numbers] Administrator
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     * (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>
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  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> 
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  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> 
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  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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     * (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>
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- ==== Sequence of Numbers ====+==== Sequence of Numbers ====
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  1. A convergent sequence has only ................ limit(s).  1. A convergent sequence has only ................ limit(s).
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     * (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>
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-  - 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) $\{\sum_{n=1}^{\infty}a_k \}$ 
-    * (D) $\{ a_n \}$  \\ <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> 
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-  ==== Series of Numbers ====+==== Series of Numbers ====
 <panel> <panel>
 1.  A series $\sum_{n=1}^\infty a_n$ is said to be convergent if the sequence $\{ s_n \}$, where .................. 1.  A series $\sum_{n=1}^\infty a_n$ is said to be convergent if the sequence $\{ s_n \}$, where ..................