How to show convergence of a sequence?

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poorform
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I have this question and other similar ones and I'm getting stuck.

Does

\displaystyle a_n=\frac{2+7n(-1)^n+n^2}{1+n^2} converge?

If so find the limit.

I am having real trouble with these types of questions I know about the sandwich theorem and trying to show \displaystyle a_n is unbounded therefore divergent but I'm very bad.

Please help.
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Slowbro93
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Have you come across algebra of limits? This may prove to be useful
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poorform
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(Original post by Slowbro93)
Have you come across algebra of limits? This may prove to be useful
\displaystyle \frac{2-7n+n^2}{1+n^2} \leqslant a_n \leqslant  \frac{2+7n+n^2}{1+n^2}

Then dividing the fractions on both sides (top and bottom) by \displaystyle n^2 and applying the AOL for sums products and quotients we get.

\displaystyle 1 \leqslant a_n \leqslant  1 and so \displaystyle a_n \rightarrow 1

Is this right?
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TeeEm
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(Original post by poorform)
I have this question and other similar ones and I'm getting stuck.

Does

\displaystyle a_n=\frac{2+7n(-1)^n+n^2}{1+n^2} converge?

If so find the limit.

I am having real trouble with these types of questions I know about the sandwich theorem and trying to show \displaystyle a_n is unbounded therefore divergent but I'm very bad.

Please help.

From the point of a "method mathematician" and not a purist

if you divide top and bottom by n2

then you get terms of O(1/n) and O(1/n2) which tend to zero as n tends to infinity

the limit is 1
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DFranklin
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(Original post by TeeEm)
From the point of a "method mathematician" and not a purist

if you divide top and bottom by n2

then you get terms of O(1/n) and O(1/n2) which tend to zero as n tends to infinity

the limit is 1
These questions are (almost) universally expected to be done from a "first principles" pure point of view, in which case that isn't going to be acceptable. (Although it would be perfectly acceptable in a pure course at postgrad level ironically).
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DFranklin
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(Original post by poorform)
\displaystyle \frac{2-7n+n^2}{1+n^2} \leqslant a_n \leqslant  \frac{2+7n+n^2}{1+n^2}

Then dividing the fractions on both sides (top and bottom) by \displaystyle n^2 and applying the AOL for sums products and quotients we get.

\displaystyle 1 \leqslant a_n \leqslant  1 and so \displaystyle a_n \rightarrow 1

Is this right?
Yes that's fine.
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TeeEm
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(Original post by DFranklin)
These questions are (almost) universally expected to be done from a "first principles" pure point of view, in which case that isn't going to be acceptable. (Although it would be perfectly acceptable in a pure course at postgrad level ironically).
I have no idea what is expected but I suspected that might have been the case.
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