Don't understand convergence as n approaches infinity

Mathguy15

Here's the deal....

I don't understand the limit as n→∞ of [(1+(.05/n))^20n -1]/[.05/n] My Calculus book says that it's supposed to approach {e^[(.05)(20)]-1}/[.05], but the numerator is a constant while the denominator goes to 0 as n→∞. The textbook, by Dr. Gilbert Strang, has similar limits, and its not in the selected errata, so I doubt it was just some error, can someone help me? Just about any light anyone could shed on this would be helpful :]

Cheers,
Mathguy

Boorglar

This looks like an indeterminate form of the type "(1^(infinity) - 1)/0", from which it is not clear whether it converges or not. You cannot be sure that it is a nonzero constant over 0, as it might be that the numerator's limit is actually 0 (and you'd get a 0/0 form).

To find this limit, you can't use direct substitution, you must use something like l'Hopital's Rule, probably twice or more (some algebraic manipulations might be required before applying l'Hopital's Rule).

Mathguy15

But the numerator is (1+(.05/n))^20n)-1, which is e^[(.05)(20)]-1=e-1 in the limit, so it has to be a constant (unless I missed something here. I just want to understand it D:)

lurflurf

That looks like a typo, what page of what edition is that on?
[(1+(.05/n))^(20n) -1]/[.05]->{e^[(.05)(20)]-1}/[.05]=20(e-1)
[(1+(.05/n))^(20n) -1]/[.05/n]->infinity