- #1
kylera
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Homework Statement
Find the limit of the sequence
{ sqrt(2), sqrt(2sqrt(2)), sqrt(2sqrt(2sqrt(2))) ... }
Homework Equations
Limit Laws?
The Attempt at a Solution
I wrote out the first five values in the sequence and came to the conclusion that this sequence could be written out as
[tex]A_{n} = 2^\frac{2^{n}-1}{2^{n}}[/tex]
I then took [tex]\frac{2^{n}-1}{2^{n}}[/tex], broke it down to [tex]1 - \frac{1}{2^{n}}[/tex] which allowed me to rewrite the equation to [tex]2\times2^\frac{-1}{2^{n}}[/tex]. Ignoring the 2 for now, I re-worked the fraction exponent and resulted with [tex]-(\frac{1}{2})^{n}[/tex] and made the value into a fraction [tex]\frac{1}{2^(\frac{1}{2})^{n}}[/tex].
Using the sheer power of what is known as the graphing calculator, I was able to determine that the limit of that equation is 1, and then multiplying 2 to it gave 2. Without a calculator, how can I lay out the steps?
Note to self: BUY A TABLET!
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