# Few problems with sequences

1. Mar 10, 2006

### rahl__

i have a few problems with sequences
1. show, that if:
$$\lim_{n\to\infty}a_{n}=L$$
than sequence:
$$b_{n}=\frac{a_{1}+...+a_{n}}{n}$$
is convergent to L

2. show that the sequence$$a_{n}$$ is monotone, bounded and find out its limit, if:
$$a_{1}=2$$
$$a_{n+1}=\frac{a_{n}+4}{2}$$

3. show that if the sequence $$a_{n}$$ satysfies cauchy's condition than it is convergent.

4. show that there is an inequility :
$$|\sum_{k=1}^{n}a_{k}b_{k}|\leq\sqrt{\sum_{k=1}^{n}a_{k}^{2}}\sqrt{\sum_{k=1}^{n}b_{k}^{2}}$$

5. find the limit of such sequence:
$$a_{n}=(\frac{n+1}{n})^{3n^{2}}$$

6. find the limit of such sequence:
$$a_{n}=(\frac{n^{2}+4}{n^{2}+3})^{2n}$$

7. find the limit of such sequence
$$a_{n}=-n^{6}+3n^{5}+7$$

8. find the limit of such sequence
$$a_{n}=\sqrt[n]{n!}$$

9. find the limit of such sequence
$$a_{n}=1+2^{n}-3^{n}$$

10. $$a_{n}$$ is a sequence including all rational numbers. show that for each real number M you can find a subsequence of this sequence that is convergent to M

11. $$a_{n}$$ is a squence, that has a subsequence convergent to $$\infty$$ and a subsequence convergent to -$$\infty$$. show that, if $$\lim_{n\to\infty}(a_{n}-a_{n-1})=0$$, than for each real number M there is a subsequence convergent to M.

thanks in advance and sorry for the length of this post, but i really need this answers as soon as possible

Last edited: Mar 10, 2006
2. Mar 10, 2006

### VietDao29

Whoops, sorry, but we are not giving out COMPLETE SOLUTIONS to those who do not even bother to try to find a way to tackle the problem(s). Why must we help him if he shows no interest in finding the solutions on this own? And to remind you, it's your own problems, not ours... :grumpy: :grumpy: :grumpy:
You may want to read the Forums Rules.

Last edited: Mar 10, 2006
3. Mar 13, 2006

### rahl__

you took me wrong, these problems were not my homework exercises. I had an exam last saturday from sequences and multitude theory [?] and from a list of about 100 exercises these 11 were the ones that i had some problems when trying to solve. i thought that this exam will be in 2 weeks time, but it turned out to be on previous saturday so i had very little time to do all those exercises and thats why i have posted just bare examples without my thoughts regarding the possible sollution, its not like im that lazy or sth.
sorry for creating such confusion

Last edited: Mar 13, 2006
4. Mar 13, 2006

### siddharth

I'm sorry rahl, but the https://www.physicsforums.com/showthread.php?t=5374", which you agreed to, says:
So, when you show your efforts, thoughts or any ideas on the problems, people like Vietdao29 and others will gladly assist you.

Last edited by a moderator: Apr 22, 2017