Re: Limit+fractional part


Patrick Coilland wrote:
eugene nous a récemment amicalement signifié :
Patrick Coilland wrote:
eugene nous a récemment amicalement signifié :
A N Niel wrote:
In article <1151671442.146228.172730@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
eugene <jane1806@xxxxxxxxxx> wrote:

Here is the limit:

limt_{n -> oo} sqrt(N) * ( 1 - max_{1 <= n <= N} ( sqrt{n} ) ) .

I've got 0 as the value of this limit. Am i right?

Thanks


Your n and N seem confused...

Yes, you are right. Sorry

Here is the limit:

limt_{N -> oo} sqrt(N) * ( 1 - max_{1 <= n <= N} ( sqrt{n} ) ) .

I've got 0 as the value of this limit. Am i right?


I don't understand :
What is "sqrt{n}" ?
Is it the same as "sqrt(n)" ?

This sqrt{n} is the same as sqrt(n), but this

sqrt{n} is in the brackets { ... } which mean the fractional part.

I have not seen these brackets : they have disappeared.


So you want to write :
L = limit_{n -> oo} sqrt(n)(1 - max_{1 <= k <= n} frac(sqrt(k)))
where frac(x) = x - floor(x) is the fractional part of x.

In such a case :

1) for x in [p^2, (p+1)^2[ :
frac(sqrt(x)) = sqrt(x) - x + 1 <= sqrt((p+1)^2-1) - p

why frac(sqrt(x)) = sqrt(x) - x + 1, frac(sqrt(x)) = sqrt(x) -
[sqrt(x)].
anyway that's no matter, i think it was a typo:

frac(sqrt(x)) = sqrt(x) -p <= sqrt((p+1)^2-1) - p , because x is
integer <(p+1)^2.


2) Let g(x) be g(x) = sqrt((x+1)^2-1) - x
g(x) is an increasing function of x

So :

g([sqrt(n)]) <= max_{1 <= k <= n} frac(sqrt(k))) < g([sqrt(n)]+1)

That's the point which i don't undertsand:

- at first, why the upper bound here is g([sqrt(n)]+1), maybe
g([sqrt(n)]) - it should be according to what you said in 1).
And i don't understand how did you get the lower boudn, which i'm most
interested in.


Could you please specify this points.

sqrt(n)(1-g([sqrt(n+1)])) < sqrt(n)(1 - max_{1 <= k <= n}
frac(sqrt(k))) <= sqrt(n)(1-g([sqrt(n)]) )

But :
sqrt(n)(1-g([sqrt(n)])) = sqrt(n)(1 - sqrt(([sqrt(n)] +1)^2-1) +
[sqrt(n)] )
= sqrt(n)/ (1 + sqrt(([sqrt(n)] +1)^2-1) +
[sqrt(n)])
and the limit of this expression is 1/2

So L = 1/2

--
Patrick


Thanks

.