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So, I'm stuck on this question, and have been working on it for a few hours, probably because I'm not 100% in understanding the definition. But here's the question:

Using the $\epsilon$ - N definition of the limit prove that,

$\lim_{x \to ∞}{1\over(n^2+n)}$ = $0$ (sorry I don't know how to write fractions in limits).

In other words, given $\epsilon$ > $0$, find explicitly a natural number N which satisfies the statement in the definition of the limit.

So the definition I received in lectures is:

Let $(a_n)_1^∞$ ($1$ should be $n=1$, I couldn't get it to work, sorry) be a sequence of real numbers $(a_n)_1^∞$ = {$a_1, a_2, a_3,...$} .

Definition: We write $\lim_{n \to ∞} a_n = a$, and say the limit of $(a_n)_1^∞$ equals $a$, if for every $\epsilon$ > 0 there exists N ∈ $\mathbb{N}$ such that if $n ≥ N$, then $|a_n - a| < \epsilon$.

So far this is what I've done:

|$\frac{1}{n^2+n}$ + 1| < $\epsilon$

$\therefore$ |$\frac{1-n^2+n}{n^2+n}$ + 1| < $\epsilon$

I did try other ways like disregarding the 'n' in the denominator since $n^2$ is more significant then it, but I don't think I'm supposed to do that. I'm really just stuck on what to do next as I don't really understand the process. Any help would be GREATLY appreciated, thanks! :)

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Let us use the definition you stated in your post. (BTW, the attempt you made is wrong).

Let $\epsilon>0$. By using the Archimedean Property, we can find $N\in\Bbb N$ such that $\frac{1}{N}<\epsilon$. Thus, if $n\geq N$ then we get $$\bigg|\frac{1}{n^2+n}-0\bigg|=\frac{1}{n^2+n}<\frac{1}{n}\leq \frac{1}{N}<\epsilon.$$ Hence, $$\lim_{n\to\infty}\frac{1}{n^2+n}=0.$$

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  • $\begingroup$ How is the substantively different from mine? ;-)) $\endgroup$ – Mark Viola Mar 15 '17 at 3:07
  • $\begingroup$ @Dr. MV Trying to avoid the function $\lfloor\cdot\rfloor$ and use Archimedean Property instead. $\endgroup$ – Juniven Mar 15 '17 at 3:09
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In the OP, it was implicitly assumed that the limit was $-1$. This occurred when writing

$$\left|\frac{1}{n^2+n}-\color{red}{(-1)}\right|<\epsilon$$

The $\color{red}{-1}$ should be $\color{blue}{0}$ instead.

To show that $\lim_{n\to \infty}\frac{1}{n^2+n}=0$ we let $\epsilon>0$ be given. Then, we have

$$\left|\frac{1}{n^2+n}-\color{blue}{0}\right|\le \frac{1}{n}<\epsilon$$

whenever $n>N(\epsilon)=\lfloor \frac{1}{\epsilon}\rfloor +1$. And we are done.

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  • $\begingroup$ Should it be that the OP was implicity assumed that the limit is $-1$? $\endgroup$ – Juniven Mar 15 '17 at 3:05
  • $\begingroup$ @ΘΣΦGenSan Yes, of course. Edited and thank you. $\endgroup$ – Mark Viola Mar 15 '17 at 3:06
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Given $\epsilon>0$ you need to find an $N$ such that $$ \left|\frac{1}{n^2+n} - 0\right| < \epsilon$$ for all $n>N.$ (The zero in the absolute value is where the limit goes. Not sure where your $+1$ came from, that would mean the limit was $-1$.)

So, you need to choose $N$ large enough so that $$ \frac{1}{n^2+n} < \epsilon$$ for all $n>N.$ (You can drop the absolute value sign since it's positive.)

We know we can do this cause the left hand side gets smaller and smaller as we take $n$ larger and larger (This is just what the limit being zero means intuitively). At a certain point $n=N$ the left hand side will become smaller than $\epsilon$ and it will be that way for all $n$ larger than that. To find the crossover point we just rearrange the inequality and get $$ n^2 +n > \frac{1}{\epsilon}$$ or $$ n^2+n-\frac{1}{\epsilon}>0.$$ For some value of $\epsilon,$ you can sketch this parabola and find the region of $n$ for which the inequality holds true. The crossover value $N$ will be the next integer after the largest solution to $n^2+n-1/\epsilon =0.$

Alternatively, we can see that if $n^2 > \frac{1}{\epsilon}$ ( or even if $n>\frac{1}{\epsilon}$) then $n^2+n > \frac{1}{\epsilon}.$ So it suffices to solve one of the much easier former inequalities. The $N$ you get that way won't be the crossover value, but it will be larger and that's all we need.

Also note that if $\epsilon$ is chosen to be particularly large (say $\epsilon>1$) Then the inequality will hold for all $n$, so you only need to consider $\epsilon$ smaller than $1.$

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  • $\begingroup$ Thanks heaps for the reply! But, I was just wondering what $n^2+n > (1$/$\epsilon$) actually tells you. Like I understand epsilon is a kind of distance from the actual limit and your trying to get within the bounds of epsilon by definition, but I just don't get what this sort of thing tells you while you're doing the calculations, my lecturer doesn't really explain that :/ Thanks again! $\endgroup$ – James Blair Mar 15 '17 at 4:25
  • $\begingroup$ @DanielConnell It is equivalent to $$ \frac{1}{n^2+n} < \epsilon.$$ This tells you that the LHS $\frac{1}{n^2+n}$ is smaller than $\epsilon.$ If $\frac{1}{n^2+n}$ goes to zero, we should be able to make it as small as we like for sufficiently large $n$. In other words we should be able to choose an $N$ to make it less than $\epsilon,$ no matter how tiny $\epsilon$ is. $\endgroup$ – spaceisdarkgreen Mar 15 '17 at 4:31
  • $\begingroup$ Ahhhh thank you so much! $\endgroup$ – James Blair Mar 15 '17 at 4:33
  • $\begingroup$ It is enough if $n^2 > 1/\epsilon$, since then certainly $n^2+n > 1/\epsilon$. This makes it much easier. $\endgroup$ – marty cohen Mar 15 '17 at 4:48

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