eze21
13.12.2019 •
Mathematics
Give examples of the following: 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
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Ответ:
1down voteAn unbounded sequence can have a convergent subsequence. An example is an=nan=n for n odd, 00for n even.The correct contrapositive of Bolzano-Weierstrass is: a sequence with no convergent subsequence is unbounded.As for (−1)nlogn(−1)nlogn, consider any real x. For n≥e|x|+1n≥e|x|+1, |(−1)nlogn−x|≥1|(−1)nlogn−x|≥1, so no subsequence of the sequence converges to x.
Ответ:
204=x addition property of equality (add 4 on both sides)
x=204 symmetric property