Why does this sequence converge to 0
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Why does this sequence converge to 0

[From: ] [author: ] [Date: 11-05-05] [Hit: ]
This is true everytime this occurs.-Because the denominator grows faster than the numerator as n increases.......
An = [2^(n-1) + 1] / [3^(n-1) + 1]

It converges to 0. Why is this?

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take out the ones for starters as the limit goes to infinity and sum constants don't effect the function (much)

An lim->inf [2^(n-1)/(3^(n-1)]

now which one grows faster 2^(n-1) or 3^(n-1)? well seeming as how the coefficient is larger in 3^(n-1) it will grow at a faster rate. So the function will go to zero because the denominator grows at a faster rate than the numerator. This is true everytime this occurs.

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Because the denominator grows faster than the numerator as "n" increases.
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