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Prove with definition that it is Cauchy

$$a_n = \frac{n+3}{2n+1},$$ wheree $n$ is a natural number

I have seen other examples such as $\frac{1}{n}$ and such that show how to prove they are Cauchy but I am confused on how to choose $N$ in this case.

szw1710
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MathMan
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2 Answers2

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We have: For any $\epsilon > 0$, choose $N > \dfrac{10}{\epsilon}$, and if $m , n \ge N$, then: $\left|a_m-a_n\right|=\dfrac{5|m-n|}{(2m+1)(2n+1)}<\dfrac{5}{2m+1}+\dfrac{5}{2n+1}< \dfrac{5}{m}+\dfrac{5}{n} \le \dfrac{5}{N}+\dfrac{5}{N}= \dfrac{10}{N}< \epsilon$. This shows $\{a_n\}$ is a Cauchy sequence.

DeepSea
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  • Thank you. So pretty much we can make anything look like 1/n since if there's other terms in the denominator it will always be less than 1/n? – MathMan Oct 14 '18 at 21:33
  • @MathMan: For this problem, yes since you have more linear terms below than above, and you can freely drop those terms you don't want. You might get $\dfrac{1}{n^2}< \epsilon$, and it still does it for $N > \sqrt{\epsilon}$. – DeepSea Oct 14 '18 at 21:38
  • ok thank you for your help! – MathMan Oct 14 '18 at 21:40
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Notice that $$\frac{n+3}{2n+1} = \frac12 + \frac5{4n+2}$$Clearly the $\frac12$ doesn't affect the "cauchiness" of the series, so this is equivalent to seeing whether $\frac5{4n+2}$ is cauchy, which is a very similar proof to that of $\frac1n$.

Rushabh Mehta
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