Practice Questions for NET JRF Real Analysis Assignment: Sequences of Real Numbers I

1. The sequence \(a_{n}=\sqrt{A^{2} n^{2}+n+1}-n\) is:

(a.) convergent for all real values of \(A\)
(b.) divergent for all real values of \(A\)
(c.) convergent for exactly one real value of \(A\)
(d.) convergent for exactly two real values of \(A\)

2. The least upper bound of the sequence \(\left\langle 1-\frac{1}{n}\right\rangle\) is:

(a.) 0
(b.) -1
(c.) 1
(d.) None of these

3. Let \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) be sequences of real numbers defined as \(a_{1}=1\) and for \(n \geq 1\), \(a_{n+1}=a_{n}+(-1)^{n} 2^{-n}\), \(b_{n}=\frac{2 a_{n+1}-a_{n}}{a_{n}}\). Then:

(a.) \(\left\{a_{n}\right\}\) converges to zero and \(\left\{b_{n}\right\}\) is a Cauchy sequence
(b.) \(\left\{a_{n}\right\}\) converges to a non-zero number and \(\left\{b_{n}\right\}\) is a Cauchy sequence
(c.) \(\left\{a_{n}\right\}\) converges to zero and \(\left\{b_{n}\right\}\) is not a convergent sequence
(d.) \(\left\{a_{n}\right\}\) converges to a non-zero number and \(\left\{b_{n}\right\}\) is not a convergent sequence

4. Which of the following sequences \(\left\langle a_{n}\right\rangle\) is monotonic?

(a.) \(\left\langle a_{n}\right\rangle=1-\frac{(-1)^{n}}{n}\)
(b.) \(a_{n+2}=\frac{1}{2}\left(a_{n}+a_{n+1}\right)\), where \(a_{1}
(c.) \(a_{n+1}=\sqrt{\frac{a b^{2}+a_{n}^{2}}{a+1}}\) for all \(n\), where \(a>0\), \(0
(d.) None of the above

5. Define a sequence \(s_{n}\) by \(s_{n}=\sum_{k=1}^{n} \frac{1}{\sqrt{n^{2}+k}}\). Then the limit of \(s_{n}\) as \(n\) tends to infinity:

(a.) is 0
(b.) is 1
(c.) is \(\infty\)
(d.) doesn't exist

6. Let \(\left\{x_{n}\right\}\) be a real sequence. If the sequence of even terms of \(\left\{x_{n}\right\}\) converges to 1 and the sequence of odd terms converges to -1. Then the sequence \(\left\{x_{n}\right\}\) will:

(a.) converge to 0
(b.) converge to 1
(c.) converge to -1
(d.) None of these

7. Let \(S_{n}=\sum_{k=2}^{n} \frac{(-1)^{k}}{k \ln k}\). Then the sequence \(\left\{S_{k}\right\}\):

(a.) converges to a finite number
(b.) diverges to \(\infty\)
(c.) diverges to \(-\infty\)
(d.) oscillates

8. Consider the interval \((-1,1)\) and a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) of elements in it. Then:

(a.) Every limit point of \(\left\{a_{n}\right\}\) is in \((-1,1)\)
(b.) Every limit point of \(\left\{a_{n}\right\}\) is in \([-1,1]\)
(c.) The limit points of \(\left\{a_{n}\right\}\) can only be in \(\{-1,0,1\}\)
(d.) The limit points of \(\left\{a_{n}\right\}\) cannot be in \(\{-1,0,1\}\)

9. Consider the sequence:

4, 0, 4.1, 0, 4.11, 0, 4.111, 0, ...

Then:

(a.) converges to \(4 \frac{1}{9}\)
(b.) is divergent
(c.) is unbounded
(d.) is not convergent and has a supremum of \(4 \frac{1}{9}\)

10. Let \(x_{n}=2^{2 n}\left(1-\cos \left(\frac{1}{2^{n}}\right)\right)\) for all \(n \in \mathbb{N}\). Then the sequence \(\left\{x_{n}\right\}\):

(a.) does NOT converge
(b.) converges to 0
(c.) converges to \(\frac{1}{2}\)
(d.) converges to \(\frac{1}{4}\)