i have a few problems with sequences(adsbygoogle = window.adsbygoogle || []).push({});

1. show, that if:

[tex]\lim_{n\to\infty}a_{n}=L[/tex]

than sequence:

[tex]b_{n}=\frac{a_{1}+...+a_{n}}{n}[/tex]

is convergent to L

2. show that the sequence[tex]a_{n}[/tex] is monotone, bounded and find out its limit, if:

[tex]a_{1}=2[/tex]

[tex]a_{n+1}=\frac{a_{n}+4}{2}[/tex]

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

4. show that there is an inequility :

[tex]|\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}}[/tex]

5. find the limit of such sequence:

[tex]a_{n}=(\frac{n+1}{n})^{3n^{2}}[/tex]

6. find the limit of such sequence:

[tex]a_{n}=(\frac{n^{2}+4}{n^{2}+3})^{2n}[/tex]

7. find the limit of such sequence

[tex]a_{n}=-n^{6}+3n^{5}+7[/tex]

8. find the limit of such sequence

[tex]a_{n}=\sqrt[n]{n!}[/tex]

9. find the limit of such sequence

[tex]a_{n}=1+2^{n}-3^{n}[/tex]

10. [tex]a_{n}[/tex] 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. [tex]a_{n}[/tex] is a squence, that has a subsequence convergent to [tex]\infty[/tex] and a subsequence convergent to -[tex]\infty[/tex]. show that, if [tex]\lim_{n\to\infty}(a_{n}-a_{n-1})=0[/tex], 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

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# Homework Help: Few problems with sequences

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