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Given that 7x2-5 (x+1)(x2+2x + 2)K X+lSx + L for some constant real x2+2x+2numbers K and L. Find the sum K + L.Enter an integer or a fully reduced fraction such as ...

Question

Given that 7x2-5 (x+1)(x2+2x + 2)K X+lSx + L for some constant real x2+2x+2numbers K and L. Find the sum K + L.Enter an integer or a fully reduced fraction such as 4 -16 3/7 1/9 etc

Given that 7x2-5 (x+1)(x2+2x + 2) K X+l Sx + L for some constant real x2+2x+2 numbers K and L. Find the sum K + L. Enter an integer or a fully reduced fraction such as 4 -16 3/7 1/9 etc



Answers

In calculus, partial fractions are used to calculate the sums of infinite series. Find the partial-fraction decomposition of the summand. $$\sum_{k=1}^{\infty} \frac{4}{k(k+1)(k+2)}$$

In this question here were given us infinite sum submission. Arm the minus one hour. Okay for the times. Gave a, uh minus one from one to infinity. Here. Want to find asked, but the best of some sq. So it's just from one up to the Jew, and now we have minus one bell. Okay, escape minus one way. We write it down. Out. We got the minus one off one, and then plus the one off to. Then we get equal to the minus. One plus one. Number two, you go to the Manus and a half. And now the s far. It will be submission from one too far. Minus one. Damn. Okay. Cable minus far. And then we should get equal to the minus. One plus one. Number two minutes. One number three and then plus one of the far. If we computers in Uganda minus seven on the trail, then last one will be asked six here because you're a submission from 1 to 6 minus one. Broke a cable minus, uh, cabe minus one. So using me one here and then it will expand and we got the ask for and then minus one number five plus one of the six here. And if we do the submission, we get the minus 37 under 60. So we got the answer for the institute. The answer for the S four and the answer for the and six.

So first we needed that identity. So six to a powerful K S t CO two. So he came minus two k injure. Here we have the two to the cane tessalit. So we mind is in a week, perhaps plus two to the K in streak A class one miners to Kate plus one. So this is because yeah. So first, it's very Lee to prove this identity. Then we know that six. Hey, ender C K pass one minus two k plus one times K minus two k. So this turn is actually Nico to minus two k pass one. Speak a minus two k last to Katie in the streak. A pass one minus two. Cape House. What? Right and we have sweet Cape class World minus two K plus one end. We came by this to Kate and this is equal to minors to cake. Last 13 cable plus one minus two K plus one plus two K industry K minus two K passport. So if we consider the sub mention here so we can see the submission. OK, you co 212 year affinity in the summiting here. Kay eager to one to your community. So? So we know that you turn up with this well, goes to zero Because sweetie to the power of Kay takes a leader, and it's under denominator. So this will go to their Oh, and this is also the telescoping sound. So what matters? Here is the first time the first thing is to to the one sweet to the white man has to know what So this is too

But we want to find the partial fraction decomposition of the summon, so we're first going to decompose it. So we have what a over K plus B over K plus three. So then we said nine equal to eight times came plus three plus B times K. And if we set K equals zero, we get nine equals three I. And if we divide by three on both sides, we get a equals three. Then if we set K equal to negative three, we get nine equals negative. Three k negative three b overnight Divide my night of three amble sides and we it b equals negative three. And if we plug those in, we get three over K minus three over K plus three as our partial fraction decomposition.

Images signify expression. Gave trying on your bunny. My gay plus two Qatari. Oh, and yeah, I noticed that I would give the same new Morita here for the denominator notice. Then we can reach. There was a K plus two Victorio que blessed you'd hams. Okay, this one times gay and then Thames game on this one of your own. Away to the Jew. Times one and so on. And something for the numerator again graduate. Domesticate him Scary man. It's one game months to on the way After the Jew Times one, we'll see that here we can against her own under term here on the top here. And I would have natural only the one honey Merica pressed you times when they pay plus one and that's gonna be the symbol form here.


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