1. ## finding an epsilon

I was wondering if someone could show me how to do this.

I need to find an N(epsilon) for the following sequences such that for all n>=N(epsilon) the absolute value of the nth element is less than epsilon >0:
1.) 1/sqrt(n)

I think I know how to do this. For e>0, 1/sqrt(n) > e. Thus, 1>sqrt(n)*e and (1/e) < sqrt(n). Square both sides and (1/e^2) < n. I believe this is an N(e) such that n >= N(e). Is this right?

2.) (1 + sqrt(n)) / (n^3)

I want to say that ((1 + sqrt(n)) / (n^3)) >= (1/n^3) > e. And then this leads to n > (1/e^(1/3)). Is this right?

3.) (sin(n)) / (2 + n^(5/3))
4.) (sqrt((n^4) +4) - n^2) * n

I'm not sure what to do on these. Any suggestions would be helpful.

2. Originally Posted by BrainMan
I was wondering if someone could show me how to do this.

I need to find an N(epsilon) for the following sequences such that for all n>=N(epsilon) the absolute value of the nth element is less than epsilon >0:
1.) 1/sqrt(n)

I think I know how to do this. For e>0, 1/sqrt(n) > e. Thus, 1>sqrt(n)*e and (1/e) < sqrt(n). Square both sides and (1/e^2) < n. I believe this is an N(e) such that n >= N(e). Is this right?
Because $\frac{1}{\sqrt{n}}$ is a strictly decreasing sequence for $n \ge 1$,
for any $\varepsilon >0$ if for some $N_{\varepsilon} \in \mathbb{N}:\ \ \frac{1}{\sqrt{N_{\varepsilon}}} \le \varepsilon$ then for all $n>N_{\varepsilon}:\ \ \ \frac{1}{\sqrt{n}} \le \varepsilon$.

So we just need to find some $N \in \mathbb{N}$ such that:

$
\frac{1}{\sqrt{N}} \le \varepsilon
$

So we solve:

$
\frac{1}{\sqrt{x}} = \varepsilon
$

and take any $N>x$ for $N_{\varepsilon}$, to make this specific we take:

$N_{\varepsilon}=\lceil x \rceil$.

RonL

3. Originally Posted by BrainMan
2.) (1 + sqrt(n)) / (n^3)

I want to say that ((1 + sqrt(n)) / (n^3)) >= (1/n^3) > e. And then this leads to n > (1/e^(1/3)). Is this right?

3.) (sin(n)) / (2 + n^(5/3))
4.) (sqrt((n^4) +4) - n^2) * n

I'm not sure what to do on these. Any suggestions would be helpful.
With all of these try finding a strictly decreasing sequence that bounds the
given sequence, then apply the method used for part one to the bounding
sequence the $N_{\varepsilon}$ you find for the bounding sequence will then suffice
for the given sequence.

By the way (4) is divergent so you will not be able to find such an $N_{\varepsilon}$.

RonL