- #1
rahl__
- 10
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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 sequencea_{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
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 sequencea_{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
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