Question

give eg..
1. An unbounded sequence that has a convergent subsequence
2. An unbounded sequence that has no convergent subsequence
3. A null sequence (an) such that the series ¡Æ an does not converge
4. A sequence which is not a Cauchy sequence but has the property that for every ¦Å > 0 and every N > 0 there exists n > N and m > 2n such that mod(an ¨C am) ¡Ü ¦Å
5. Two sequences (an) and (bn) such that the sequence (cn) defined by cn = an + bn converges to 1 but neither (an) nor (bn) converge.
6. A sequence (an) that tends to +ve infinity but is neither increasing nor eventually inc

Answer 1

here the whole question comes
Give examples of:
1. An unbounded sequence that has a convergent subsequence
2. An unbounded sequence that has no convergent subsequence
3. A null sequence (an) such that the series ¡Æ an does not converge
4. A sequence which is not a Cauchy sequence but has the property that for every ¦Å > 0 and every N > 0 there exists n > N and m > 2n such that mod(an ¨C am) ¡Ü ¦Å
5. Two sequences (an) and (bn) such that the sequence (cn) defined by cn = an + bn converges to 1 but neither (an) nor (bn) converge.
6. A sequence (an) that tends to +ve infinity but is neither increasing nor eventually increasing.
7. A bounded sequence (an) such that (an+1/an) tends to 1 but (an) doesn¡¯t converge.
8. A bounded sequence which doesn¡¯t converge.
9. A series ¡Æ an which converges but such that ¡Æ 3¡Ìan diverges.
10. A series ¡Æ an with sequence of partial sums (sn) such that sn = an for all natural numbers n.
11. A sequence (an) of non-zero terms such that (an+2/an) tends to 1 but (an) does not converge.
12. A sequence (an) of non-zero terms which is bounded above but such that the sequence (1/an) is not bounded below.
13. A convergent series ¡Æ an with all terms non-zero and ¡Æ an ¡Ù 7 which can be rearranged so that its sum is 7.
14. A series ¡Æ an which converges but such that ¡Æ (-1)^n+1.an does not converge.
15. A sequence (an) such that, for all natural numbers m, an = m for infinitely many natural numbers n.
16. A Cauchy sequence
17. A sequence of the form (x^n + y^n)^(1/n) which converges to 6, as n tends to infinity.
18. Two series ¡Æ an and ¡Æ bn which diverge to infinity but where ¡Æ an.bn converges.
19. A sequence (an) which satisfies the following but does not diverge to infinity:
for all C > 0, there exists N so that there exists n > N with an > C
20. A sequence (an) with (1/an) tending to 0 but (an) not tending to infinity.
21. A non-monotonic, unbounded sequence.
22. A sequence (an) which satisfies the following but does not converge to 1:
for all ¦Å > 0, for all N there exists n > N with mod(an ¨C 1) < ¦Å
23. A number which has only one decimal expansion
24. A sequence of irrationals which has a subsequence converging to 2 and a subsequence converging to 3.
25. A sequence (an) with (an+1/an) tending to 1 but which does not converge.
26. A series ¡Æ an, with non-zero terms, which can be rearranged so that its sum is 3.5.
27. A series ¡Æ an, with (an) tending to 0 and with (sn) tending to infinity , where (sn) is the sequence of partial sums.
28. A sequence (an) which tends to infinity but has an+1 < an for infinitely many n.

Answer 2

1. An unbounded sequence that has a convergent subsequence
\[ a_{2n}=2n\quad a_{2n+1}=\frac 1{n+1}\]

2. An unbounded sequence that has no convergent subsequence
\[ a_{n}=n\]

3. A null sequence (an) such that the series ¡Æ an does not converge
It is not clear what you want(unreadable characters)

4.A sequence which is not a Cauchy sequence but has the property that for every ¦Å > 0 and every N > 0 there exists n > N and m > 2n such that mod(an ¨C am) ¡Ü ¦Å
It is not clear what you want (unreadable characters)

5. Two sequences (an) and (bn) such that the sequence (cn) defined by cn = an + bn converges to 1 but neither (an) nor (bn) converge.
\[ a_{n}=n\quad b_n=-n+1 \]

6. A sequence (an) that tends to +ve infinity but is neither increasing nor eventually increasing.
\[ a_{n}=\frac n 2 +\sin(n) \]
7. A bounded sequence (an) such that (an+1/an) tends to 1 but (an) doesn¡¯t converge.
\[ a_{n}=n \]

8. A bounded sequence which doesn¡¯t converge.
\[ a_{n}=sin(n) \]


9. A series ¡Æ an which converges but such that ¡Æ 3¡Ìan diverges.
It is not clear what you want(unreadable characters)


10. A series ¡Æ an with sequence of partial sums (sn) such that sn = an for all natural numbers n.

\[ a_{n}=0 \]

11. A sequence (an) of non-zero terms such that (an+2/an) tends to 1 but (an) does not converge.
\[ a_{n}=n \]
12. A sequence (an) of non-zero terms which is bounded above but such that the sequence (1/an) is not bounded below.
\[ a_{n}=\sin(n) \]

13. A convergent series ¡Æ an with all terms non-zero and ¡Æ an ¡Ù
7 which can be rearranged so that its sum is 7.
It is not clear what you want(unreadable characters)
14. A series ¡Æ an which converges but such that ¡Æ (-1)^n+1.an does not converge.

\[ a_n =(-1)^{n+1} \frac 1 n\]

15. A sequence (an) such that, for all natural numbers m, an = m for infinitely many natural numbers n.
\[ a_{n,m}= n \text { for all n and m} \]
It is an infinite matrix whose nth row is eqaul to n

16. A Cauchy sequence
\[ a_n =\frac 1 n\]

17. A sequence of the form (x^n + y^n)^(1/n) which converges to 6, as n tends to infinity.
\[ x=0 \quad y= 6 \quad (x^n + y^n)^{1/n} = 6 \]
for all n.

18. Two series ¡Æ an and ¡Æ bn which diverge to infinity but where ¡Æ an.bn converges.
\[ a_n = b_n = \frac 1 n \]

19. A sequence (an) which satisfies the following but does not diverge to infinity: for all C > 0, there exists N so that there exists n > N with an > C

\[ a_{2n}=0\quad a_{2n+1}=2n+1\]


20. A sequence (an) with (1/an) tending to 0 but (an) not tending to infinity.

\[ a_n = (-1)^n n \] This sequence does not tend to [\ +\infty\]


You can do the rest.