Consider the sequence

and define,

.

n,k positive natural numbers.

Find:

Using induction I found that

But I don't know how to find the limit.

The limit should be

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- Dec 22nd 2009, 06:45 AMm3th0dmanLimit of sequence
Consider the sequence

and define,

.

n,k positive natural numbers.

Find:

Using induction I found that

But I don't know how to find the limit.

The limit should be - Dec 22nd 2009, 07:37 AMtonio

According to what you wrote:

, so:

and etc...., so I think you may have not written what you meant...

Tonio - Dec 22nd 2009, 07:42 AMm3th0dman
- Dec 22nd 2009, 10:39 AMtonio
- Dec 22nd 2009, 12:20 PMLaurent
I have been taught to avoid L'Hospital' rule, so here is what I wrote:

and that's it.

Moreover, the same method works for this general problem. Let us write for simplicity, and , for all .

Let . Start from:

.

The definition of is just an intricate way to describe the coefficients in this expansion.

Indeed,

,hence ,

,hence ,

and so on, "unfolding the expansion", until

,hence .

If you really want to write every word of the proof, then you have to explicitate the above induction. Let and prove by induction on that :

(you could also use convergent series instead of asymptotic expansions to shorten the proof, but these would be a little inappropriate here since we only care for the limit) - Dec 23rd 2009, 01:13 AMm3th0dmanThanks
Thank you very much!