Use a CAS to perform the following steps for the sequences in Exercises 137-148 . a. Calculate a

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

Key Concepts

Nth Root Convergence Property

For sequences defined by taking the nth root of a fixed positive number, it is a standard result that as n increases, the nth root approaches 1. This property arises from the nature of roots and logarithms, and understanding it is crucial when investigating sequences of the form a^(1/n).

Epsilon Criterion for Convergence

The epsilon criterion is a formal definition of convergence that states for every positive tolerance (often designated as epsilon), there exists an integer N such that for all indices n greater than or equal to N, the absolute difference between the nth term of the sequence and the limit is less than epsilon. This idea is used to quantitatively describe how close the sequence terms come to the limit.

Limits

The limit of a sequence is the value that the terms approach as the index goes to infinity. Calculating and understanding limits is central to the study of sequences, and various techniques can be applied—analytical or numerical—to determine this behavior.

Computer Algebra Systems (CAS)

A Computer Algebra System (CAS) is a software tool that facilitates performing complex algebraic computations symbolically or numerically. In the context of sequences, a CAS can be used to compute terms, generate plots, and help in verifying properties such as convergence and boundedness.

Boundedness

A bounded sequence is one whose terms stay within fixed upper and lower limits. Determining whether a sequence is bounded involves checking if there exist finite numbers M and m such that every term of the sequence lies between m and M. This property is important when investigating convergence.

Sequence

A sequence is an ordered list of numbers generated by a specific rule or formula. Understanding sequences involves identifying how each term is produced, analyzing its behavior as the index increases, and studying properties like monotonicity, boundedness, and convergence.

Convergence

Convergence of a sequence means that as the sequence progresses to infinity, the terms approach a specific value known as the limit. Analyzing convergence involves verifying that the distance between the terms and the limit becomes arbitrarily small beyond a certain index.

Transcript

00:01 For this problem, we are asked to use a cas to first calculate and then plot the first 25 terms of the sequence, a -n equals the nth root of n.

00:10 We are asked, does the sequence appear to be bounded from above or below? does it appear to converge or diverge? and if it does converge, what is the limit, l? so i have that established right here.

00:20 We have n to the power of 1 over n.

00:21 Then i generate a table of values for n between 1 and 25.

00:26 And then i plot that data.

00:27 So we can see that output 115 shown here is the result of the first, or calculating the first 25 terms.

00:36 We can see that it appears to be bounded below by one and also bounded above by this value of about 1 .44 there.

00:44 It appears to be converging, and specifically it appears to be converging to a value of, with the current angle, it seems to be about maybe 1 .13, but i'll actually conclude that, it is really going to be converging to a value of 1.

01:02 It does say to only plot the first 25 terms of the sequence, but if we expand that out to the first 50, we can see that it goes down below, even 1 .1, or if we expand that out to the first 100, we can see that it continues approaching one point, or it continues approaching 1, just very slowly.

01:21 So we can conclude that l is probably going to equal 1.

01:26 Then we are asked in part b, if the sequence converges, find an integer n such that the absolute value of a .n minus l is less than or equal to 0 .01 for a little n greater than or equal to big n.

01:39 And how far in the sequence do you have to go for the terms to lie within 0 .0 ,0001 of l? now what i'll do is generate that, actually, i'll take a slightly different approach here.

01:52 I'll do a discrete plot of a n, or actually absolute value of as well as the value 0 .01, the value 0 .00, we want 3 .0 .0 .0 .0 .00 .0 .1.

02:17 So, one second here.

02:22 Oh, that was silly.

02:24 And i'll plot this out for n between 1 and 25 to start.

02:29 So we can actually see that if we're only considering between 1 and 25, it would appear that it doesn't look like it's convermed...

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