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| Both sides previous revision Previous revision Next revision | Previous revision | ||
| msc:mcqs_short_questions:real_analysis [2023/04/01 18:37] – [Edit - Panel] 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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| * (D) does not exist \\ <btn type=" | * (D) does not exist \\ <btn type=" | ||
| </ | </ | ||
| - | ==== Sequence of Numbers ==== | + | ==== Sequence of Numbers ==== |
| < | < | ||
| 1. A convergent sequence has only ................ limit(s). | 1. A convergent sequence has only ................ limit(s). | ||
| Line 104: | Line 107: | ||
| * (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|< | ||
| Line 164: | Line 169: | ||
| * (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) $\{\sum_{k=1}^{\infty}a_k \}$ | ||
| - | * (B) $\{\sum_{k=1}^{n}a_k \}$ | ||
| - | * (C) $\{\sum_{n=1}^{\infty}a_k \}$ | ||
| - | * (D) $\{ a_n \}$ \\ <btn type=" | ||
| </ | </ | ||
| - | ==== Series of Numbers ==== | + | ==== Series of Numbers ==== |
| < | < | ||
| 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 .................. | ||
| Line 234: | Line 233: | ||
| * (C) $\sum |a_n|$ is convergent. | * (C) $\sum |a_n|$ is convergent. | ||
| * (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent. \\ <btn type=" | * (D) $\sum |a_n|$ is divergent but $\sum a_n$ is convergent. \\ <btn type=" | ||
| + | </ | ||
| + | 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) $\{\sum_{n=1}^{\infty}a_k \}$ | ||
| + | * (D) $\{ a_n \}$ \\ <btn type=" | ||
| </ | </ | ||
| ==== Limit of functions ==== | ==== Limit of functions ==== | ||
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| * (C) $3.8$ | * (C) $3.8$ | ||
| * (D) $0.1$ \\ <btn type=" | * (D) $0.1$ \\ <btn type=" | ||
| - | < | + | </panel> |