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0/1 points Previous Answers LarCalc11 9.2.031 Find the sum of the convergent series.24n(n + 2) n =Need Help?Read ItITelk to 4 MutorSubmit AnswerSave ProgressPractic...

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0/1 points Previous Answers LarCalc11 9.2.031 Find the sum of the convergent series.24n(n + 2) n =Need Help?Read ItITelk to 4 MutorSubmit AnswerSave ProgressPractice Another Versior

0/1 points Previous Answers LarCalc11 9.2.031 Find the sum of the convergent series. 24 n(n + 2) n = Need Help? Read It ITelk to 4 Mutor Submit Answer Save Progress Practice Another Versior



Answers

Find the sum of the convergent series. $$\sBum_{n=1}^{\infty} \frac{4}{n(n+2)}$$

Hello. We have some here is the sum and going from one to infinity of eight divided by M plus one times M plus two. So we can bring an eight out front of the summer. And we have eight times the sum and go from one to infinity of one over M plus one minus one over N plus two, which then gives us eight times um times what times a one half minus one third plus 1/3 -1 4th. Um and so on here and a plus uh 1 4th um minus 1/5 right and so on. Um And that is going to be equal to, well it's going to be equal to just eight times one half. So therefore all we get here is just eight times one half and the sum is equal to four.

Okay. We want to evaluate the sun off this Siri's. So, first of all, we used to hint we use partial fraction. We can read this as 1/2 times one over end, minus one over and plus two. Okay, so on, this is equal to we can put the 1/2 in the front some and from one to affinity, One over and minus one over and plus two. Okay, so if you write out this Siri's So this is what you're looking at is the one that's one minus one over three and plus one over two, minus one over fool. Plus one over three, minus one over five, etcetera, etcetera. And, as you can see here, except one and 1/2 everything. For example, 13 will get cancelled by the one over three. Here. If you further arrived out the next time one of our four minus one over six, then you will see this one over four got canceled with this one over four. Right? So everything else will get cancelled eventually, except one and 1/2. So this is in fact, 1/2 times one plus 1/2. Okay, So the answer is three over for

To break down with these two Siri's subtract thio. We could take the Siri's individually and then subtract those. And so each of these is going to be a geometric. The first one is gonna some too well, I've just rewritten. 0.4 is 4/10 and 0.8 is 8/10. The first one is going to some 21 over one minus 4/10 and then the next one. So then we have subtract, and then we've got one over one minus 8/10 and then we just need to break these down. And so the one on the left here we dio in the denominator one minus 4/10. That'll be 6/10. But then flip that that'll be 10 6 and then the next one is one divided by two tents. And so that'll flipped Turn over to then we might as well just multiply this one top and bottom by three. Get a common denominator. So that'll end up giving us 30 six and then finally subtracting these two. Now we'll end up getting, of course, the 10 6 and then minus 36. So that's where we'll get negative. 26th, but of course, we could break that down, divided to these by two, so there'd be negative 10 thirds. So that would be our final. Some here when we some all of these terms and tracked on them and what not?

This is a sequence where you might just recognize that each time we're moving from one term to the next or multiplying by negative one half and then we multiply again by negative one half for each step here. If you didn't see that, one way to do it is to just take a subsequent term like negative two and divide back by the previous term. And that would also get us this multiplication tive rule, which is negative one half. So you could always divide this by this and that will also get us with that multiplication is. And so we could write this as a geometric series since its multiplied each time by the same amount, and it starts off with four. So that's gonna be the value that will put in the front there. And then we put our ratio are which is, of course, negative. One half to the power. And and just to verify that this expression works here, if we plug in and a zero, we get the first term and is one. We get a second term, and it's too. We get the third term and so on and then to jump right toe. What the some adds up to you starts off with whatever the coefficient is divided by one minus What's in parentheses? That our value one half the minus minus tells us, Of course, we're just gonna plus thes. So then this turns into four divided by three halves over, really have. And then that two flips back up to the top, and then that ends. It ends up giving us eight thirds. So as we add these terms out to infinity here, then they're going to some thio.


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