In Problem 6 you dealt with sequences of the type {(1+(1)/(an))^bn} where both an and bn diverge

In Problem 6 you dealt with sequences of the type...

In Problem 6 you dealt with sequences of the type $\left\{\left(1+\frac{1}{a_n}\right)^{b_n}\right\}$ where both $a_n$ and $b_n$ diverged to infinity. In the first part of Problem 6, $a_n$ and $b_n$ diverged to infinity at the same rate; in the second part, they diverged to infinity at different rates. Try to find a precise connection between the limit of the sequence and the relative rates at which $a_n$ and $b_n$ diverge to infinity.

Key Concepts

Limit Forms of Exponential Sequences

This concept involves understanding sequences of the form (1 + c/n)^n and their convergence to the constant e as n approaches infinity. It generalizes to sequences where both the additive term and the exponent vary, emphasizing that when the ratio between the exponent and the denominator of the base argument stabilizes, the sequence converges to e raised to the limit of that ratio.

Relative Growth Rates

Relative growth rates are central to comparing how two sequences diverge to infinity. In this context, the limit of (1 + 1/a_n)^(b_n) is determined by the asymptotic behavior of the ratio b_n/a_n. If b_n grows at the same rate as a_n, the limit tends to e, but if one grows faster than the other, the limit will be e raised to the ratio of their growth rates, illustrating the sensitivity of limits to the relative divergence of the sequences.

Logarithmic Transformation and Asymptotic Analysis

This key idea uses the logarithm to transform the sequence into a form that is easier to analyze by turning exponentiation into multiplication. Approximating log(1 + 1/a_n) as 1/a_n for large a_n simplifies the expression to b_n/a_n. This asymptotic analysis highlights how the behavior of the sequence is governed by the limiting behavior of the ratio of the sequences, thereby connecting the limit directly to the relative rates of divergence.

Transcript

00:02 All right, so we've got a bunch of statements about limits and what happens when you perform operations on them, basically.

00:13 So part a says the limit as n goes to infinity of a .n is one.

00:20 And the limit as n goes to infinity of bn is three.

00:23 Then the limit as n goes to infinity of bn over a .n is three.

00:29 And this is going to be true.

00:31 If both the limits exist in the top and bottom, then you can take the limit separately and then evaluate as long as the bottom limit isn't zero.

00:39 Okay, so that's good.

00:42 Maybe we'll keep track here.

00:44 So a is true.

00:47 Okay, part b, now we've changed it.

00:49 Limit for a .n is zero, and the limit for bn is infinity.

00:56 And they're now asking about the limit of a .n.

00:59 Times bn.

01:01 And we're claiming that that is zero.

01:06 Well, that's not true.

01:08 We don't know if that's shrinking faster than that's growing.

01:16 Okay, so a counter example.

01:21 Here's an example that shows it's not true.

01:23 Let's say a .n equals 1 over n and bn equals n.

01:30 Then a .n times bn is 1.

01:32 So the limit is n goes to infinity of a and bn is not zero, it's one.

01:39 Okay, part c, we have two convergence sequences, a and bn, and they differ in their first hundred terms, but an equals bn after 100.

01:50 So then the limit as n goes to infinity of both of them are equal, and that's true, because the limit does not care about the first hundred terms at all.

02:01 Okay, part d, a .n is 1 over n, basically...

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