This is probably really obvious, but I am not seeing it.

Why does,

?

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- Apr 14th 2009, 07:49 PMmollymcf2009series convergence question
This is probably really obvious, but I am not seeing it.

Why does,

? - Apr 14th 2009, 08:34 PMChris L T521
Apply L'Hopital's rule. By direct evaluation, it has the indeterminate form of

Let

We then see that . Now if we directly evaluate the limit, we have the case , which then satisfies the condition to apply L'Hopital's Rule.

Now,

But,

Therefore,

In a more general sense, I leave it for you to show that

Does this demystify things? - Apr 14th 2009, 10:36 PMchisigma
Just a little observation: the use of L'Hopital rule in demonstration that...

(1)

... presupposes that you have demonstrated before that...

(2)

... and that presupposes you have demonstrated before that...

(3)

... and that presupposes you have demonstrated before that...

(4)

... and that presupposes you have demonstrated before (1)...

So invoking L'Hopital's rule for demostration of (1) is a sort of 'vicious circle' and is better to avoid it...

Kind regards

- Apr 14th 2009, 11:43 PMNonCommAlg
you can't prove because it's just a definition! (Giggle) well, unless you have another definition of and you want to prove that the one you have implies the one that we have! (Nod)

anyway, the only thing which needs to be proved here is that exists, which it does of course. now is just a name for the limit. - Apr 15th 2009, 12:23 AMchisigma
It is interesting however the demonstration that the number e can be defined as limit in two different ways...

From the pratical point of view the accurate computation of e as...

... requires an n of the order of several milions, while the accurate computation of e as...

... requires n less than 10! (Wink)...

Kind regards