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Hi.
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In this video, we are going to use the root test to determine whether this series, the sum from n equals 2 to infinity of n over the natural log of n to the nth power converges or diverges.
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So to use the root test, we need to evaluate this limit.
00:23
The limit as n goes to infinity of the terms of the series, absolute valued and raised to the 1 over n power.
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So, well, everything's positive here, so we can drop the absolute values.
00:44
So let's do that.
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And then we'll bring the 1 over n into the top and bottom.
00:51
So this will leave us with n to the 1 over n divided by the natural log of n.
00:59
So here, we know that as n goes to infinity, the natural log of n is going to go to infinity.
01:11
Right? so the denominator is going to grow and grow and grow.
01:15
But what about n to the 1 over n? what happens to this as end goes to infinity? that's not immediately clear, but we do have some ways of evaluating or finding this limit.
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So let's focus on just on this limit here.
01:37
So the limit, i'll do this in three, the limit as n goes to infinity of n to the 1.
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Over n.
01:46
So the first thing we're going to do is use properties of exponentials and natural algorithms to rewrite this as e to the natural algorithm, uh, n to the one over n.
02:03
Right...