Determine the limit of the sequence: an = (1+(5/n))2n
L'hopitals rule, or at least that's what I'm thinking. Otherwise, general formulas for determining the limit of a sequence.
The Attempt at a Solution
an = (1+(5/n))2n
Considering the behavior of the sequence as n goes toward infinity, (5/n) is a very small change in size, so it goes approximately to zero, leaving:
an = (1)2n
Which, as n goes to infinity, is essentially: 1∞, an indeterminate form. To resolve this, take the natural log:
an = ln(12n) = 2n(ln(1))
Rearranging it to give a L'hopital's rule acceptable form of 0/0, gives:
And taking the derivative:
0/(-1/2n2) = 0
And raising to e to get back to the original value:
e0 = 1
Except this is clearly incorrect, as we determined in class that the sequence converted to something around 22,000, and not 1.
Other than this, I tried not canceling the (5/n), taking the log, and then splitting it using the limit law:
lim[f(x)*g(x)] = lim(f(x)) * lim(g(x))
But this didn't work either, as the limit of 2n by itself diverges as n goes to infinity, and this sequence is known not to diverge.