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JH
Numerade Educator

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Problem 5 Medium Difficulty

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4 + n^2} $

Answer

It appears that the series is convergent.

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Video Transcript

let's find the first aid terms of the sequence of partial sums. So the sequence is denoted in your books by sn here. We're starting at one, and we go up to infinity. So we'd like to find the first eight terms here. So that's s one all the way to s eight. And then we'll try to see if this series is conversion or diversion. So as one by definition, is just a one and this is our a m right here. So a one is just one over one to the fourth plus one squared one half. You can write that as 0.5, that will be our decimal. And then as to we can write, this is s one plus a two. In general, we can always write S n is equal to s n minus one plus an so s one we know that's just one half from over here. And then we add in a two So a two and then we'll go ahead and simplify this. So actually I've using Wolfram Alpha. I've actually computed all eight partial sums here, and these are the fraction. So this one half is what we had in part one. So continuing in this fashion, this is s one as to s three, 4567 and eight. However, and this answer, they want four decimal places. So all do here is just rewrite the decimals that I get from Wolfram. So, for example, for us to we go to the second row. Here we have 11 half 11/20. Excuse me. That's what we would get after simplifying this summer. Fractions. Over here you have 11/20 and that is just 0.5500 That's four decimals again, and we'll just keep going all the way until we get to s eight. As three. This is as two plus a three. So s two plus one over three to the fourth plus three squared and you would go over to the Wolfram. We see that's 10 1/1 80 and then let's go ahead and round off their That's approximately 0.5611 again, we're going to four decimal places. So that's why we stopped there. So that s three. Now, let's go to s four in a different color as four would be s three, which we just found. Plus a four, which is one over for the fourth plus four squared. We would go ahead and simplify that. Or use Wolfram and using Wolfram 6913. 12,240 the denominator. And then just round that off using a calculator. 24 places. That's what I get. 240.5648 Similarly, for s five. So that's s four. Plus a five or s four plus 1/5 to the four plus five squared. Right. So here we can take a peek at the Wolfram. We're on the fifth term here, so we do have this large numerator and denominator in row five. So that numerator 450,000 5. 69 that's all on the top. And then in the denominator, 795,600. And this is S five here. And when we round this off using the calculator, this is approximately 50.5663 So we have five down, three more to go, so I'll go on to the next page since I'm running out of room here. Uh huh. So this will be as six and then using all from this is our fraction. So here we have 16 million, 693,000. 153 up top in the new writer and the denominator. 29 million 4. 37 and then 200. And let's go ahead and round that off. And if you run this off, this is Rs six, and this is approximately using a calculator. I get 60.5671 We have two more to go. Here's S seven. That s six plus a seven. Right. So that's the previous answer, Essex. Plus this. So here I am not rounding off. This s six that I'm using is not the decimal. This is the exact value that we got over here. And this. I've been doing this method the whole time. All of my s values are firm. This wolfram table here, these are all exact values and for the approximation of approximating each one individually. So for s seven. Go ahead and simplify this or use the wolfram. 818 million. So all of this is in our numerator In the denominator were there in the billions. There's 44242 to 800. And once again use that calculator to round off their This is 5675 approximately. That's going to four places. And for the last one s eight, using the same method we've been using all along. So use Wolf from here. We get three billion and some change in the numerator. A lot of change there and then in the denominator five billion and some change. And if we round this off using a calculator, that's 0.5677 That's one of four places again, and it appears that the sequence will converge. It looks like we're converging to around 5.567 ish, maybe eight. Here, let's we can also Well, you can't see it from here because this is fraction form. Let me exit this. So let me go to the next page. So we had s one. As to all the way to s A and we started with 0.5 0.55 0.5611 0.5648 0.5663 0.5671 0.5675 and then 0.5677 So looking at these numbers and it appears that it will converge to this is just we're not supposed to actually give the exact answer here. We were supposed to say whether it appears that it converges. I'm saying that it will, and more or less it will be about 0.5, say 568 And there's our final answer.