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Hello there.
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In the following exercise we need to determine if the following series that is equal to 2 to the n power divided by 3 to the n power minus 1, converge or diverge.
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To do this we are going to use the limit comparison test.
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That basically what they say is that if you have two series and you have some information from one of this to series you need to take the sequences the limit.
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It to the end, tending to infinite of the sequences.
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And if this converts to some value that is positive, then we can say that this two series behave in the same way.
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That means that either both converge or both diverge.
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So in this case, we need to determine to which series we can compare.
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So it is preferable to check what is going to happen in the infinity.
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So in the infinity we're going to have two dominant terms that is two to the n power and three to the n power.
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So in that sense, we can compare this to the series given by two -thirds to the n power which we know that is a convergent series because it's a geometric series of the form a r to the n power a n, yes, here, a, n is a, r to the n power...
