- #1

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## Homework Statement

Determine the limit of the sequence: a

_{n}= (1+(5/n))

^{2n}

## Homework Equations

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

a

_{n}= (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:

a

_{n}= (1)

^{2n}

Which, as n goes to infinity, is essentially: 1

^{∞}, an indeterminate form. To resolve this, take the natural log:

a

_{n}= ln(1

^{2n}) = 2n(ln(1))

Rearranging it to give a L'hopital's rule acceptable form of 0/0, gives:

ln(1)/[1/2n]

And taking the derivative:

0/(-1/2n

^{2}) = 0

And raising to e to get back to the original value:

e

^{0}= 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.