For my homework is say to prove that you are correct and that you may quote theorems. I am unsure about what accumulation points are. thank you!!
An example of a sequence that has no accumulation points
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sequences-and-series
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3Do you not have a textbook to tell you what accumulation points are? Was no definition given in lectures? It's unusual for someone to assign homework, without telling the students the definitions of the terms used in the questions. – 2012-10-23
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2This is the tenth time you've posted a question directly from your homework without explaining what you've tried and without making partial progress. First, academic norms are that you should cite your sources. Are these questions taken from a textbook, if so which one? Are they written by a specific professor, if so who? Second, you seem to be getting stuck right at the beginning of many questions, and it seems like you should talk to someone in person (your prof. or TA) about how to approach these sorts of problems. – 2012-10-23
2 Answers
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The sequence $(-1)^n$ has not accumulation point Because the values are between -1 and 1 and it doesn't have limit.
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2The *set* $(-1)^n$ has no accumulation point, but the *sequence* $(-1)^n$ has two, namely $1$ and $-1$, at least with the most common definition. – 2012-10-23
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0But in the link i left say that "An accumulation point is a point which is the limit of a sequence" then (-1)^n has two limits which is impossible because the limit must be unique. – 2012-10-23
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0So does it have two accumulation points? being 1 and -1? or what would be an example of sequence with only 2 points? – 2012-10-24
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0I think the definition at MathWorld is not the standard definition. Also, contrary to mrf's comment I think with the standard definitions there's no difference between an accumulation point of a set and of a sequence. But clearly there are some different definitions floating around out there, so you should let us know what definition you are using. This would be easier if you told us what textbook you took this question from. – 2012-10-24
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0The definition in my text defines A.P. as: A point p is an A.P. of a set S if it is the limit of a sequence of points of S-{p}. – 2012-10-24
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1@Maximiliano: Ah, sorry, mrf was totally right, the definition for set and sequence should be different. The point is that sequences have repeats, while sets don't. This matters because when you "remove p" from the set you're actually removing infinitely many things in the sequence! The usual definition of accumulation point of a sequence is that it's a limit of some (infinite) subsequence. With that definition $(-1)^n$ would have two accumulation points: $1$ and $-1$. But you should double check with your text whether they distinguish between accumulation points for series and sets. – 2012-10-25
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Take $a_n = n$, and work from there.