Which of the sequences {an} converge, and which diverge? Find the limit of each convergent seque

Which of the sequences {an} converge, and which diverge?...

Key Concepts

Sequences and Convergence

This concept involves determining whether the terms of a sequence approach a finite limit as the index goes to infinity. A convergent sequence has a well-defined limit, while a divergent sequence does not settle at any particular value. Understanding the criteria and tests for convergence is fundamental in analysis.

Growth Rates and Dominance

In the context of sequence limits, it is important to recognize that certain functions grow faster than others. Logarithmic functions, even when raised to high powers, increase at a significantly slower rate compared to polynomial or linear functions. When comparing terms in a sequence, if the denominator grows much faster than the numerator, the sequence will typically converge to zero.

l'Hospital's Rule

This is a method used to evaluate limits of indeterminate forms such as 0/0 or ?/? by differentiating the numerator and denominator. By applying l'Hospital's Rule, one can often simplify the limit evaluation process and rigorously determine the behavior of a sequence as its index approaches infinity.

Transcript

00:01 We want to determine whether the sequence am is equal to natural log of n to the 200 power over n converges or diverges, and if it converges, what it converges to.

00:13 Well, let's go ahead and first just see what happens as we evaluate this at infinity.

00:20 So as in the cruxed infinity, well, we know natural log of infinity and then raising something to composite power, which is still good infinity.

00:29 So we have infinity and n goes to infinity as well.

00:33 So what this tells us is we should apply local tall rule.

00:37 So let's go ahead and take the limit of this.

00:40 So the limit as an approaches infinity of am, this is going to equal to the limit as in approaches infinity.

00:51 And this is by localities.

00:54 When we take the derivative of our numerator, so we're going to need to use chain rule to do this.

01:00 So the derivative of our outside function, well, that's x to the 200 power to use power rule.

01:07 It be 200 times the natural log of n.

01:11 Now to the 199 power.

01:14 And then we have to multiply by the derivative on the inside, which is natural log of n.

01:20 So we would multiply this by one over n.

01:23 And then we divide this by what we get in the denominator, which is 1.

01:28 Well, we can leave that one over in into the denominator, and we would be left with the limit as an approach of infinity of 200 natural log of n over 199 over in.

01:48 And if we were to look at the limit as an approach's infinity of this expression here, well, it's going to be the exact same thing that we had to start.

01:59 So it's still going to be infinity over infinity, so we still need to use lopetal's rule.

02:04 So let's go ahead and apply lopatol rule one more time.

02:08 So i'll move this down here...

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