Understanding L'Hopitals Problem Approximation

[sparkle123]

Homework Statement

How did this approximation come about? It doesn't seem like it's by L'Hopital's rule. Thanks!

Homework Equations

The Attempt at a Solution

 

[christoff]

Actually, it is L'Hopital's rule. Note that the limit might only exist if i and N have the same sign, or i is negative and N is positive (so as to guarantee we actually have an indeterminate form). We then have three cases; i<N, i=N, i>N.

Assuming the above limit exists (that is we're in the situation I mentioned above) applying l'Hopital's rule gives you\lim_{k\rightarrow ∞}\frac{1-r^{ki}}{1-r^{kN}}=\lim_{k\rightarrow ∞}\frac{-ir^{ki}}{-Nr^{kN}}=\lim_{k\rightarrow∞}\frac{i}{N}r^{k(i-N)}.
What happens with the various cases? Compare to the value of the limit \lim_{k\rightarrow ∞}\frac{r^{ki}}{r^{kN}}.

 

[Ray Vickson]

Homework Statement

How did this approximation come about? It doesn't seem like it's by L'Hopital's rule. Thanks!

Homework Equations

The Attempt at a Solution

The result is false if |r| < 1 because in that case the LHS is 1 but the RHS could be 0, infinity or 1, depending on the values of i and N. For |r| > 1 it follows easily and has nothing at all to do with l'Hospital (that is not l'Hopital---you need an 's' in it).

 

[christoff]

The result is false if |r| < 1 because in that case the LHS is 1 but the RHS could be 0, infinity or 1, depending on the values of i and N. For |r| > 1 it follows easily and has nothing at all to do with l'Hospital (that is not l'Hopital---you need an 's' in it).

True, true; forgot about |r|. Although Ray, I'm a bit confused as to why you say that this has nothing to do with L'Hopital's rule.

 

[Ray Vickson]

True, true; forgot about |r|. Although Ray, I'm a bit confused as to why you say that this has nothing to do with L'Hopital's rule.

l'HoSpital's rule is not needed, because all we need is to note that r^m - 1 = r^m * [1 - (1/r^m)] and 1/r^m --> 1 as m --> inf.

 

[christoff]

l'HoSpital's rule is not needed, because all we need is to note that r^m - 1 = r^m * [1 - (1/r^m)] and 1/r^m --> 1 as m --> inf.

Ah, true. Thanks for the clarification. And I really don't like including the s in L'Hôpital; personally I think dropping the accent and adding an s just makes the pronunciation even more confusing to newcomers. Don't want people to think he was a medical doctor or something

 

[Ray Vickson]

Ah, true. Thanks for the clarification. And I really don't like including the s in L'Hôpital; personally I think dropping the accent and adding an s just makes the pronunciation even more confusing to newcomers. Don't want people to think he was a medical doctor or something

Well his name was ... with an 's', but pronounced with no 's'. I do not feel personally empowered to alter the spelling for the sake of convenience, but that's just me.

 

[christoff]

Well his name was ... with an 's', but pronounced with no 's'. I do not feel personally empowered to alter the spelling for the sake of convenience, but that's just me.

Having just looked up the history of his name, I now agree with you. I was under the impression that the English spelling L'Hospital came from the direction translation of Hôpital. Apparently he was named L'Hospital originally, with the silent s, but then the French changed their orthography in the mid 18th century and the "os" became a ô. History defeats me again.

 

[sparkle123]

Thank you to both! :D