19-22 Calculate, to four decimal places, the first ten terms of the sequence and use them to plo

19-22 Calculate, to four decimal places, the first ten terms...

Key Concepts

Rational Functions and Asymptotic Behavior

When a sequence is defined by a rational function—one polynomial divided by another—the limit as the index tends to infinity is often determined by the leading coefficients and degrees of the polynomials. This simplification helps analyze the asymptotic behavior, showing whether the sequence converges and, if so, to what value.

Convergence and Limit of Sequences

A sequence is said to have a limit if the terms approach a specific value as the index increases without bound. Convergence is a foundational concept in the study of sequences, as it informs us about stability and asymptotic behavior. Determining the limit is crucial in many areas of analysis and calculus.

Sequences

A sequence is an ordered list of numbers defined by a specific rule. In mathematical analysis, sequences help us understand how values change and behave as the index increases, and they form the basis for studying convergence, divergence, and limits. They are central to many concepts in calculus and analysis.

Term Calculation in Sequences

Calculating individual terms in a sequence involves substituting successive integer values into the given rule. This process not only provides specific numerical instances but also helps illustrate emerging patterns that may be indicative of the overall behavior of the sequence.

Graphical Analysis of Sequences

Graphing a sequence involves plotting each term as a discrete point on a coordinate system, with the index on one axis and the term’s value on the other. This visual representation can reveal trends, such as convergence or divergence, and offers an intuitive understanding of the sequence’s long-term behavior.

Transcript

00:01 So students in this question we are given with the sequence and we need to find the first 10 terms of the sequence and also we have to plot the graph of the sequence and after that we have to tell that does the sequence appear to have a limit or not with explanation okay, so the given sequence we have a n is equals to 3n divide by a 1 plus 6 n so let's find the first 10 terms here i have made one table this is we have can write here n as 1 2 3 4 5 6 7 8 9 10 and here this is a n now i'll be putting the value of n as 1 here to find a n so when n is 1 this would become a 1 so that would be 3 divide by 1 plus 6 is 7 so that will be 0 .4286 so this is 0 .4286 now when n becomes 2 this would be 2 here so 3 times 2 is 6 and 1 plus 6 times 2 is 13 so 6 over 13 so this value comes out to be as 0 .4 615 so that would be 0 .4 615 so that would be 0 .4 615.

01:34 Clearly we can put n as 3 here and we can find a 3 and so on so i'm just writing the values so here this will be 0 .4737 then 0 .4800 when n is 5 we have a n to be 0 .4 4839 next we have 0 .4865 then 0 .4865 then 0 .4486 then 0 .4 884 then 0 .4 898 then 0 .4909 and finally we have 0 .44918 so we have obtained all the values now we can plot the graph so along the x -axis will take n and along the y -axis will take a n so this is our graph this is a n and here this is n now when n is 1 here this is n is 1 here a n is 0 .4 86 which is somewhat here 0 .428 now when n is 2 a n is 0 .4615 so 0 .4615 is here and so on we have plotted all the points now if you see from the graph when n approaches to infinite a n becomes equals to 0 .5 from the graph we can tell this thing now let's verify the results.

03:10 So we have limit in approaches to infinite.

03:17 A and we have 3n divide by 1 plus 6 n.

03:22 Now we can divide with n in the numerator and in the denominator...

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