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

Asymptotic Analysis

This involves analyzing the behavior of functions and sequences as the variable grows very large. By focusing on the dominant terms in the expression, one can approximate the limit. In sequences involving polynomials and radicals, factoring out the highest powers of the variable is a common method to determine the eventual behavior as the index approaches infinity.

Algebraic Manipulation and Rationalization

Often, sequences defined with radical expressions are simplified using algebraic techniques such as rationalization (multiplying by the conjugate). This technique helps transform the sequence into a form where the limit can be more easily evaluated by cancelling out the problematic terms.

Sequence Convergence

This concept involves determining whether the terms of a sequence approach a specific value as the index goes to infinity. A convergent sequence has a limit, while a divergent sequence does not settle at any finite or well-defined value. In analysis, testing convergence is essential to understanding the long-term behavior of sequences.

Limit of a Sequence

The limit of a sequence is the value that the sequence's terms get arbitrarily close to as the index increases without bound. Calculating limits often requires algebraic manipulation to reveal the dominant behavior of the sequence, especially when the expression is complex or involves radicals.

Transcript

00:01 We want to determine whether the sequence a -n is equal to n minus the square root of n squared minus n converges or diverges, and if it converges what it converges to.

00:14 So first, let's just see if we can evaluate this directly.

00:17 Well, as n approaches infinity, well, n goes to infinity minus the square root, and then n -squared minus n, just using some of the n behavior that we've learned about, goes to infinity.

00:31 So that would be infinity minus infinity and i don't know what that's supposed to make.

00:35 So we need to find some clever way to rewrite this into something we do know how to evaluate.

00:43 So what i'm going to do is multiply this, the top and bottom, by its conjugate.

00:51 So we're going to have n minus a square root of n squared minus n and then we're going to multiply the top and bottom by n plus the square root of n squared minus n and then we're going to multiply the top and bottom by n plus the square root of n squared minus n and then.

01:04 Minus n and divide by n plus the square root of n squared minus n and so this is equal to a n and now if we were to go ahead and boil off the top since that's the difference of squares we would get n squared minus n squared minus n all over n plus the square root of m squared minus n and then the n squared minus n squared comes out and distributing that negative to negative in we get n over n plus the square root of n squared minus now this here is in the form of something that looks like a rational sequence so there is a trick that we would do, which would be to divide the top and bottom of this by the largest power in either the numerator or the denominator.

02:17 So let's go ahead and write out this.

02:22 And now the largest power in the numerator and the denominator is just in, because we can think of the largest power in the denominator as either n or the square root of n -square, which is also just n...

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