5-8 Calculate, to four decimal places, the first ten terms of the sequence and use them to plot

step 1 Hello and welcome. We are looking at chapter 2, section 1. This is problem 5. So we're asked to

5-8 Calculate, to four decimal places, the first ten terms...

Key Concepts

Sequence

A sequence is an ordered list of numbers defined by a specific formula or rule. Studying sequences helps in exploring patterns, behaviors, and the progression of values as the terms increase, which is fundamental in calculus and analysis.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index becomes infinitely large. Determining the limit is essential for understanding the long?term behavior of sequences, whether they converge to a particular value or diverge.

Algebraic Simplification

Algebraic simplification involves transforming expressions into simpler forms by factoring, cancelling common terms, or combining like terms. This is crucial when analyzing sequences, as it often reveals properties such as the limit or rate of growth.

Numerical Approximation

Numerical approximation is the process of calculating the values of terms to a specified degree of precision which, in this case, involves computing the first ten terms to four decimal places. This technique helps in verifying theoretical insights by providing concrete data points.

Graphing Sequences

Graphing sequences involves plotting the computed terms on a coordinate plane, typically with the term index on the horizontal axis and the sequence values on the vertical axis. This visual representation is important for observing trends, convergence, or divergence in the sequence's behavior.

Transcript

0:00 Hello and welcome.

00:01 We are looking at chapter 2, section 1.

00:07 This is problem 5.

00:09 So we're asked to calculate the first 10 terms of a sequence, plot the graph of the sequence, and then kind of make some inferences about whether or not it has a limit.

00:19 If it has a limit, we need to calculate it.

00:22 So here is where we're starting at.

00:33 So i'm going to calculate the first two, and then i'll show you a table of the rest of them.

00:37 So the first one, a sub 1, we just replace all the ends with ones.

00:46 So this would be 1 squared over 2 times 1 plus 3 times 1 squared.

00:54 It's going to be 1 over 2 plus 3 or a 5th.

01:02 So that rounds to 0 .2.

01:05 And if we wanted to go to 4 decimal places, we could leave it like this, or we could put three zeros after it.

01:13 So a sub two would be similar.

01:17 Replacing all the ends here with twos.

01:23 We have two squared over two times two plus three times two squared.

01:33 That's four over four plus two squared is four times three is 12.

01:39 So order of operations there.

01:41 You have four over 16.

01:44 That's one -fourth as a decimal that is 0 .25 and we can add two zeros after it if we wanted to show that it's to the fourth decimal place extra clear so this is a sub 1 here here's a sub 2 we want to find the first 10 terms so i'm not going to bore you by calculating all 10 but i will show you a graph so this is from desmos so we can see that are equation here is the same one that we that we were given we can see that we got the same result for the first two terms and then we can see rounding to four i trust you know how to round but so this seven here for example would stay there this seven would stay as a seven this one would stay as a one it doesn't look like that rounds up this would round up here, this six would round up.

02:52 So pay attention to that if you're checking your work and so on.

02:58 So these are all supposed to be four decimal places.

03:02 This one right here is the one that needs rounded up.

03:05 Anyways, so you can look at all 10 terms there to make sure you didn't make any silly mistakes.

03:12 And we also have the graph here.

03:13 So two things to notice about the graph.

03:18 One is, well, i guess there's a lot of this.

03:20 Really only one main thing to notice about the graph is that it seems to be it's kind of growing faster and then it kind of flattens off right so that pattern where it flattens off that's strong evidence that there is a limit so this i'd say yes this sequence appears to have a limit right so then it asks us if we think it has a limit to calculate the limit so looking back our sequence we want to see as n goes to infinity does this sequence go to infinity or does it converge to something not infinity so this is the setting up our limit so whatever's in here is the sequence a sub n and so the trick we use is we find the highest power in the denominator so we have n squared this trick is when we have a polynomial over a polynomial and we look at the denominator here the highest power of n that's in the denominator, and we divide the top and bottom of our limit by n squared.

05:07 So every term in the numerator, every term in the denominator gets multiplied by 1 over n squared.

05:13 It's just another way of saying dividing it by n squared.

05:16 So this becomes n squared over n squared, all over 2n over n squared, plus 3n squared over n squared now, that's the trick...

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