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CoudTTONA CONVERGENCE WEIRDNESS: SAME NUMBERS DIFFERENT SuMs Wc will nov show that two diflercnt rcarrangemcnts of tho altcrnating harmonic scrics add up two differ...

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CoudTTONA CONVERGENCE WEIRDNESS: SAME NUMBERS DIFFERENT SuMs Wc will nov show that two diflercnt rcarrangemcnts of tho altcrnating harmonic scrics add up two different valueConsider2 4 =1-1+1-1+3 In thc PrcLab you shorcd that this scrics convcrgcd by satisfying the hypotheses of the Alternating Serics Test, Thus, may Use the Remainder in Alternating Scrics (Thcorem 20 In thc tcxt) Thcorcm to bound thc actual sumNote that the first partinl sumThe Remalnder in Alternating Serics Thcorem stated tha

CoudTTONA CONVERGENCE WEIRDNESS: SAME NUMBERS DIFFERENT SuMs Wc will nov show that two diflercnt rcarrangemcnts of tho altcrnating harmonic scrics add up two different value Consider 2 4 =1-1+1-1+3 In thc PrcLab you shorcd that this scrics convcrgcd by satisfying the hypotheses of the Alternating Serics Test, Thus, may Use the Remainder in Alternating Scrics (Thcorem 20 In thc tcxt) Thcorcm to bound thc actual sum Note that the first partinl sum The Remalnder in Alternating Serics Thcorem stated that the actual sum (of which wc will not dctermina thc eract value here) satisfies |S Sil Using propcrtics of absolutc valucs (Iii) Adding Sq = (iv) Wc conclude 5 Wc rcpeat the steps for Sz the actual sum gatisfies |S Sz| Using properties of absolute valucs <5 -52 < (iii) So, <5 < (iv) We conclude $ <



Answers

Assume as known the (true) fact that the alternating harmonic series
(1) $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\cdots$
is convergent, and denote its sum by $s$ . Rearrange the series (1) as follows:
(2) $1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\frac{1}{9}+\frac{1}{11}-\frac{1}{6}+\cdots$.
Assume as known the (true) fact that the series $(2)$ is also convergent, and denote its sum by $S .$ Denote by $s_{k}, S_{k}$ the $k$ th partial sum of the series $(1)$ and $(2),$ respectively. Prove the following statements.
(i) $S_{3 n}=s_{4 n}+\frac{1}{2} s_{2 n}, \quad$ (ii) $S \neq s$

Problem. 82 have toe proven identity between the pressure sums off the automated Syria in a rearrangement of the until, maybe seniors. So if we observed the rearrangement of the of the night, it's Siri's. We have that The sum is made of groups off three elements, one less one sir minus one have won over five won over seven, minus one over four. If we can't groups off three elements, thes one would be the first group. This one will be the second, and this one will be the end. I believe that the, uh, part hard part of this exercise is to come out with the general form to come out with this expression for in on. And I encourage you to think about how to obtain this expression. One thing that you can serve Easter if we compare one in 1/5 we are having for in the bottom over here and the same is gonna happen if we do one more, that will be nine. Okay. So that the U. S. A hint off before the three. We can obtain it because that's what we need to make one. Okay. And it's off course, very clear that between this one and this one, that is to off difference over here saying And over here. Same. So that gave us the clue off how how to generalize this summer. Um, And then we have to work with they. I'm on it serious, until for Orin. So that's that's that approximation. And we are gonna also use one cup that sum to bury this time until to it. So what I did here is just multiply expression that it will be 4 to 1. This one is No. That's hard to come out. Okay, so what I have to prove now East, these two, if we had them best. So let's up the two expressions. This the same that we have a gulp? No, I wanna take away from the 1st 1 All the ones with even bottom, and I'm gonna separate them. He missed them over here. Why? Because the off bottom? Oh, yes. Correspond to these terms. Okay, The only ones were missing at this one. This one and this one, And those ones were gonna obtain it from this part. So the next thing tops our Eastern here. We have negative 1/2 here we have a positive one. Have 1/4 if this doesn't work because they have the same sign. But it's gonna work for 16 He's gonna work for every number four even number. That is no Phoebe. Civil by four. Okay, so we can get rid of this one. And this one, we will get rid of seats. You have that? They're here and we will get Reid. Oh, these two. Okay. So what do we have? We have that expression here. We have for a love, love, love, love until four. Okay, but this is now multiplied by two. Because we have it two times. So if we were here and multiply by two each element inside the sun, we'll get minus one car minus one. For now, we have all the even numbers, and we see that the vin numbers that we have here I'm the same that we required over there. So let's rearrange the Siri's again. And we're hoping the same expression that we haven't up. We have think s three a. And this is what we have to do for party now, for part B, we have to use this expression to check that this sum is, in general, Not the same for they automated harmonic serious in this re arrange other made a demonic Sears. So suppose that the limit of their Monix serious the S is Yeah. I shall have to use a connection. Um, and we want a check. How much is the limit of the rearrange has donated harmonics years. So is the same The calculatedly bit off a scent and took up a little bit of history. And if the live it exists is gonna be guessing. Now we can use six press option that we find. So we have the limit when it goes to the media there? Yes. Or fall in Plus 1/2. What is off to it? No, this is ill. And this is to buy the sim property that we use election. So we have l less 1/2 of l this 3/2 Seville and it's not the same. Therefore the sounds are modest

In this problem. The A M If you can't you and tube and first fi and force CASS Tree So again, If been to the forced fei into the forest sweet, we know it's converging to finance non their number one verify When end goes to infinity. This means by the emergence task there, Siri's die mergers.

In this exercise, we have to apply Test given in the exercise. 35 to the Termini. Why Dysfunction guy Burgess. So the best in the exercise 30 part says that we have a serie well, general term that day in we know it verges If when we calculate the dreamy Oh, in a N that is a final number that it's not Cyril. So we're gonna cooperate. The name it. We're in Khost infinity off end times and cute over five times into the four plastic. This is the same that Inter for over five into four plus three. So what we're gonna do is take that into the four to the bottom so we'll have five into the four divided by ending for last three, divided by into the fourth. Now the first term in the denominator we met them. We can simplify into the four. And in the second term, we can't simply by, but this goes to Syria when then goes to Philip. So when it goes to infinity, we only get one over five and this is a final number on it's not zero. Therefore, by the criteria, we know that the Siri's die Burgess


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