Use a CAS to perform the following steps for the sequences....

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L$ ? b. If the sequence converges, find an integer $N$ such that $\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$ $$ a_{n}=\left(1+\frac{0.5}{n}\right)^{n} $$

Use a CAS to perform the following steps for the sequences.
a. Calculate and then plot the first 25 terms of the sequence.
Does the sequence appear to be bounded from above or
below? Does it appear to converge or diverge? If it does
converge, what is the limit $L$ ?
b. If the sequence converges, find an integer $N$ such that
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\left(1+\frac{0.5}{n}\right)^{n}
$$

Key Concepts

Sequence Behavior

This concept involves analyzing a sequence of numbers by observing its terms. It includes generating a finite number of values (in this problem, the first 25 terms) and using visual or numerical techniques to understand its trends such as increasing or decreasing tendencies.

Limit of a Sequence

Determining the limit of a sequence means finding the value that the sequence approaches as the index goes to infinity. In the context of the given sequence, recognizing that the form (1 + constant/n)^n converges to an exponential limit is crucial, which generalizes to the well-known limit definition of e raised to the constant.

Boundedness

A sequence is said to be bounded if there exists a number that serves as an upper or lower limit beyond which the sequence does not go. Understanding boundedness helps determine whether the sequence stays within finite bounds, a key property for convergence analysis.

Exponential Function and e

This concept refers to the exponential function and its properties, particularly the mathematical constant e. The sequence (1 + 0.5/n)^n is an example that converges to e raised to a power (in this case e^(0.5)), illustrating the fundamental limit definition of e in calculus.

Error Estimation and Tolerance

Error estimation involves determining the point in the sequence where the difference between the sequence term and its limit falls below a specified threshold. In this problem, the goal is to find an index N such that the absolute difference |a_n - L| is less than or equal to given tolerances (0.01 and 0.0001), which is a common task in numerical approximation and convergence criteria.

Transcript

00:01 We want to use some kind of technology to determine the first 25 terms of the sequence.

00:06 Am is equal to 0 .9999 to the nth power.

00:10 And we also want to determine whether the sequence appears to be found from above or below, as well as whether it appears to converge or diverge.

00:19 And if it does converge, we need to say what we think the limit should be.

00:23 So that's for part a.

00:24 And then if it does converge to something, we can do some things for part b.

00:29 So let's go ahead and answer to part a first.

00:32 So looking at this chart here, so i used excel to generate this.

00:38 Our first column is just what term of the sequence it is, and a -n is our output for that term.

00:46 So you can see that these values really don't stray too far away from 0 .99.

00:56 Nine.

01:00 So it looks like it's monotonic and it's decreasing.

01:05 So just from looking at the first 25 terms of the sequence, it looks like it will be convergent to something, but just using these numbers, we can't really tell at least the first 25 terms.

01:21 So a .n.

01:24 Converges, or at least it kind of looks like it's converging because we know it's monotonic.

01:28 And this also looks like it is bounded above by one since it's monotonically decreasing.

01:39 And we know raising a positive number to a positive power will always be bigger than zero, so we can at least say this is bounded between zero end...

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