Which of the sequences in Exercises 115-124 converge, and which diverge? Give reasons for your a

Which of the sequences in Exercises 115-124 converge, and...

Key Concepts

Convergence of Sequences

This concept refers to the behavior of a sequence that approaches a specific limit as the index increases indefinitely. A sequence converges if, beyond some point, all its terms remain arbitrarily close to a particular number, the limit. Understanding convergence is critical for analyzing the long-term behavior of sequences.

Divergence of Sequences

Divergence occurs when a sequence fails to approach any fixed number, meaning no matter how far out you go, the terms do not settle near a single value. Recognizing divergence is important, as it shows that a sequence does not have predictable limiting behavior.

Limit Calculation Techniques

This involves methods and algebraic manipulations used to evaluate the limit of a sequence. By applying limit laws and simplifying expressions—often by isolating dominant terms or using properties of fractions—one can determine if a sequence converges and what value it converges to.

Behavior of Exponential Functions

Exponential functions, depending on their base, can exhibit rapid growth or decay. For instance, understanding that the reciprocal of an exponentially growing function tends to zero as the index increases is crucial when evaluating limits in sequences. This insight helps in simplifying sequences where exponential terms are involved.

Transcript

00:01 So to determine if this converges or diverges, the first thing i would do is let's distribute this 3 to the end.

00:13 So we're going to look at instead 2 to the n over 3 to the end minus 1 over 3 to the end.

00:20 And then let's pull out the end from each of these.

00:24 So this is going to be 2 over 3, raise to the n, minus.

00:29 And even if it's not written, 1 is also kind of raised to the power end because 1 to the end is always just going to be 1.

00:39 And so this here would also be 1 3rd raised the end.

00:44 So this is just rewriting this for us.

00:48 Because notice if we were to just try to apply this limit directly, so as n goes to infinity.

00:55 Up here we get 2 to the infinity minus 1 over 3 to the infinity and then 2 to the infinity, 3 to infinity are both infinity so this just ends up being infinity over infinity so it's like i don't know what that means but if we do this rewriting here so we still have where this is like 2 thirds to infinity minus 1 3rd to the infinity but we actually have a couple of rules for this that will help us up so let's take the limit now as an approach to infinity of this so we have a and now i'm just going to squeeze in the limit as an approach to infinity on each side here.

01:33 Like that.

01:35 So the first rule we can use is the sum or difference rule.

01:40 So assuming both of these limits exist, we can rewrite this as the limit as an approach is infinity of two -thirds raise to the end minus the limit as an approaches infinity of one -third raised to the end...

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