Limits

[FUGU]

lim (1ⁿ+2ⁿ+..+nⁿ)/n^​n+1
n->∞

 

[Deleted member 4993]

limit (1ⁿ+2ⁿ+...+nⁿ)/nⁿ⁺¹<br>
n->pozitive infinity

Does your problem look like:

\(\displaystyle \displaystyle{\lim_{n \to \infty}\frac{1^n + 2^n + .... + n^n}{n^{n+1}}}\)

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

 

[FUGU]

Does your problem look like:

\(\displaystyle \displaystyle{\lim_{n \to \infty}\frac{1^n + 2^n + .... + n^n}{n^{n+1}}}\)

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

yes , it does! in the preview of the post it looked like this.Thought it was sort of a bug..

 

[FUGU]

lim (1ⁿ+2ⁿ+..+nⁿ)/n^​n+1
n->∞
I dont know if i shall be using lema stolz sezaro, bacause i was told that it should be done like this.Any help?

 

[Deleted member 4993]

yes , it does! in the preview of the post it looked like this.Thought it was sort of a bug..

So where is your work?

What do you need to do? - show the limit exists? calculate the value of the limit? what..?

What are the theorems that you know that may be applicable to this problem?

 

[Ishuda]

lim (1ⁿ+2ⁿ+..+nⁿ)/n^​n+1
n->∞
I dont know if i shall be using lema stolz sezaro, bacause i was told that it should be done like this.Any help?

Would it do any good to re-write the series as
\(\displaystyle S_n\, =\, \frac{1}{n}\underset{j=0} {\overset{n-1}{\Sigma}}\,\,(1-\frac{j}{n})^n\)
and note that
\(\displaystyle \underset{n\, \to\, \infty}{lim}\, (1+\frac{x}{n})^n\)
has an upper bound depending only on x?

Actually I'm not sure if that would do any good or not, but it is the way I would start to tackle it.

 

[FUGU]

it is actually a really good idea,as you can write it as an euler number exponential now. It was hard to find the general term for the sum.The only problem now is to calculate the limit of the exponent

 

[Ishuda]

it is actually a really good idea,as you can write it as an euler number exponential now. It was hard to find the general term for the sum.The only problem now is to calculate the limit of the exponent

Suppose that upper bound was t^j for a fixed t then what would nS_n look like? Also is there a 'good' upper bound on that, that is something that grows slower than n [or both an upper and lower bound that grow the same as n]?

 

[FUGU]

Suppose that upper bound was t^j for a fixed t then what would nS_n look like? Also is there a 'good' upper bound on that, that is something that grows slower than n [or both an upper and lower bound that grow the same as n]?

so ive thought of this[(1-k/n)^-n/k]^(-k/n)*n .From here it results e ^-kFrom here you may continue the sum.That's what i think...

 

[Ishuda]

so ive thought of this[(1-k/n)^-n/k]^(-k/n)*n .From here it results e ^-kFrom here you may continue the sum.That's what i think...

If t<1
\(\displaystyle \overset{n}{\underset{j=0}{\Sigma}}\, t^j\, =\, \frac{1-t^{n+1}}{1\, -\, t}\,\, \le\,\, \frac{1}{1-t}\)