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

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$ 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 $\left|a_{n}-L\right| \leq 0.01$ 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 in Exercises $129-140 .$
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 $\left|a_{n}-L\right| \leq 0.01$ 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 Convergence and Divergence

This concept involves determining whether a sequence approaches a specific value (converges) or fails to settle around any particular value (diverges) as the index increases indefinitely. It is central to understanding the long-term behavior of sequences and is linked with the ideas of series, limits, and stability in mathematical analysis.

Boundedness of Sequences

Boundedness characterizes whether the terms of a sequence stay within a finite upper and lower limit. A bounded sequence does not exceed certain thresholds, which is critical in convergence tests and in ensuring that certain computational methods (like those using Computer Algebra Systems) remain stable and reliable.

Limits of Sequences

The limit of a sequence is the value that the sequence approaches as the index goes to infinity. This concept is crucial when analyzing sequences, determining convergence, and understanding asymptotic behavior. In many instances, the limit serves as a foundational idea for further work in calculus and real analysis.

Utilizing Computer Algebra Systems (CAS) for Sequence Analysis

CAS tools are employed to calculate, plot, and analyze sequences efficiently. They facilitate the computation of many terms, the graphical visualization of sequence behavior, and precise investigation into features such as boundedness, convergence, and error margins. This assists in handling complex or iterative sequence computations with high accuracy.

Rate of Convergence and Error Tolerance

This concept addresses how quickly a convergent sequence gets arbitrarily close to its limit. It involves quantifying the number of terms needed for the sequence to be within a specified distance (error tolerance) from the limit. This is particularly important when performing numerical approximations and when using computational tools to assess convergence.

Transcript

00:01 So the sequence we were looking at today is a sub n equals 1 plus 0 .5n, sorry, 1 plus 0 .5 over n, all raise to the power n.

00:14 Excuse me.

00:16 And so today i'll be using geogrebrae calculus calculator, to evaluate and analyze the sequence.

00:24 If you're using your own calciculator, the documentation for functions may be different.

00:29 So you might not be able to do exactly what i do here.

00:34 But nevertheless, so first we have to plot the first 25 terms.

00:38 So let's see.

00:40 We're going to label l1.

00:44 And so for the sequence function, first i'm just going to get the whole numbers that we can, so we can plot the sequence against it.

00:51 So our function will be n, and variable n, starting at 1 and ending at 25.

00:58 So as you can see, this gives us a sequence of the whole numbers.

01:02 Now to get our actual sequence, once again use the sequence function with our expression.

01:15 This is the formula for the sequence in the variable n, start terms, or starting at n equals 1, ending at n equals 25, and that gives us our sequence here.

01:28 And then finally we have to plot the two together to get it on our graph, and here it is.

01:37 So you can see that it starts off at 1 .5, then it starts getting a little bigger, bigger.

01:46 And then it's hard to tell, but it does look like it is still strictly increasing.

01:53 So this is a non -decreasing sequence.

01:56 And for this non -decreasing sequence to converge, it has to be bounded from above.

02:03 So we can try to find the limit using our cast calculator.

02:09 So let's plug in the expression.

02:14 Expression.

02:28 In n, we want to evaluate the limit as it goes to infinity.

02:32 We can see that it converges to a nice square root of e.

02:37 This is the decimal form if you don't need it.

02:40 So, we know that our upper limit is square root of e.

02:45 That is our, and so the sequence does converge.

02:49 And so now we have to do some limit analysis.

02:54 Sorry, go over this.

02:58 So we need to find integer n big n such that absolute value of a n minus l so this is the distance of the sequence to the limit the distance between those two and we want to find that this distance is less than a nicer sign less than or equal to 0 .01 4 n is greater than or equal to big so what this is saying is if we have our sequence goes along like this and we have our limit.

03:46 Let me use some different colors here.

03:49 So this is our sequence...

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