Re: limit of a sequence
- From: The Qurqirish Dragon <qurqirishd@xxxxxxx>
- Date: Sat, 07 Jul 2007 06:00:43 -0700
On Jul 7, 8:01 am, "Armando C." <xxxxxxxx...@xxxxxxxxx> wrote:
I've a doubt about the following sequence limit, I tried many times
to solve it, but I couldn't work it out.
lim n->+inf ( 1/(n^2+n+1) + 2/(n^2+n+2) + ..... + n/(n^2+n+n))
My book says result is 1/2, but frankly speaking I don't understand
how it comes to it. If each term is infinitesimal how can sum be
different from 0?
Thanks in advance.
The sum is bounded from below by:
lim (n->inf) sum(i=1..n, i/(n^2+2n)) [why? I leave this to you].
Using the formula for the sum of the numbers from 1 to n, it is easy
to show that this limit is 1/2. So your given limit/sum is at least
1/2.
A similar line of reasoning will show you that it is also bounded from
above by 1/2 (I'll leave it to you to figure out the details of this.)
By the Squeeze theorem, therefore, the limit is 1/2.
Note that, more generally, this is an explicit example that the sum of
infinitesimals doesn't have to be 0.
.
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