111). The set of real numbers R together with two symbols \(+\infty\) and \(-\infty\) is called
| A). extended real number system R |
| B). complex numbers |
| C). cantorset |
| D). none of these |
|
112). If \(\{a_{n}\}\) converges to l, then the swquence \(\{x_{n}\}\) where \(\Large x_{n}=\frac{a_{1}+a_{2}+...+a_{n}}{n}\)
| A). diverges to \(+\infty\) |
| B). also converges to \(l\) |
| C). also converges to 0 |
| D). diverges to \(-\infty\) |
|
113). Discuss the convergence of the sequence, \(\Large -1,\frac{1}{2},-\frac{1}{3},\frac{1}{4},-\frac{1}{5},\frac{1}{6},...\)
| A). converges to 1 |
| B). converges to 0 |
| C). diverges to \(+\infty\) |
| D). diverges to \(-\infty\) |
|
114). Discuss the convergence of the sequence \(a^{n}=1+(-l)^{n}\)
| A). converges to O |
| B). diverges to \(-\infty\) |
| C). oscillates finitely |
| D). none of these |
|
115). Find \(\Large \lim\limits_{x\rightarrow \infty}\frac{1}{n} \left(1+2^{1/2}+3^{1/3}+....+n^{1/n}\right) \)
| A). 0 |
| B). 1 |
| C). \(\infty\) |
| D). none of these |
|
116). Find \(\Large \lim\limits_{x\rightarrow \infty} \left(\frac{n^{n}}{n!}\right)^{1/n} \)
| A). 1 |
| B). 3 |
| C). 0 |
| D). 1/e |
|
117). If \(\{ a_{n} \}\) and \(\{ b_{n} \}\) are sequences such that \(a_{n}\rightarrow +\infty\) and \(\{ b_{n} \}\) converges then \(\{ a_{n}b_{n} \}\)
| A). converges to 0 |
| B). diverges to \(+\infty\) |
| C). converges to 1 |
| D). none of these |
|
118). If the sequence \(\{ a_{n} \}\) is bounded and the sequence \(\{ b_{n} \}\) converges to zero then the sequence \(\{ a_{n}b_{n} \}\)
| A). diverges to \(+\infty\) |
| B). diverges to \(-\infty\) |
| C). converges to zero |
| D). none of these |
|
119). Find \(\lim\limits_{n\rightarrow \infty}\sup\ S_{n}\) where \(S_{n}=(-l)^{n}(n\in I)\)
| A). 1 |
| B). 0 |
| C). -1 |
| D). none of these |
|
120). If \(\{S_{n}\}_{n=1}^{\infty}\) is a sequence of real numbers that is not bounded above then
| A). \(\Large \lim\limits_{n\rightarrow \infty}supS_{n}=-\infty\) |
| B). \(\Large \lim\limits_{n\rightarrow \infty}supS_{n}=\infty\) |
| C). \(\Large \lim\limits_{n\rightarrow \infty}infS_{n}=\infty\) |
| D). \(\Large \lim\limits_{n\rightarrow \infty}infS_{n}=-\infty\) |
|
