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Prove thie following lemia: Lerma Let {Ta}i sequence of real nurnbers We define C) new sequences {E,k_] and {O-}=1 Vn € 2+ I Vn € 2 Tin-1IF the scquene...

Question

Prove thie following lemia: Lerma Let {Ta}i sequence of real nurnbers We define C) new sequences {E,k_] and {O-}=1 Vn € 2+ I Vn € 2 Tin-1IF the scquenees (E,} =1and {O};=1 #re both convergent tothe same litnit, THEN the scquence {Ia}i 1 also convergent: Suygeston: Use the definition o Iitit,

Prove thie following lemia: Lerma Let {Ta}i sequence of real nurnbers We define C) new sequences {E,k_] and {O-}=1 Vn € 2+ I Vn € 2 Tin-1 IF the scquenees (E,} =1and {O};=1 #re both convergent tothe same litnit, THEN the scquence {Ia}i 1 also convergent: Suygeston: Use the definition o Iitit,



Answers

Show that if $a_{n}>0$ and $\Sigma a_{n}$ is convergent, then
$\Sigma \ln \left(1+a_{n}\right)$ is convergent.

Suppose a N has a limit that is l and let Absalon be greater than equal to zero and pick. And so that her lower case and greater than capital end a n minus l is less. Seminary is less than Absalon, divided by two. Thus for n greater than began. Excuse me. I need to back up here just one second for N and M greater than capital. We have a M minus. A M can be re written using l. And now let's go ahead and use triangle inequality to separate them. And this is less than absolute. Over to an absolute over to buy construction, which is I've flown over two plus absolute over two is equal to Absalon. This finishes approved.

Were given that the Siri's is conversion and that the terms in the syriza positive. Let's show that this Siri's also converges. So here, let's use the Lim comparison test. So this requires that we look at the limit as n goes to infinity of the natural log one plus a M over this other term over here, which is just an now, let's go ahead and replace a N with Let's Do X equals one over and we know that the limit of a n equal zero. That's just using the divergence test since this Siri's convergence. So by diversion test. So since X equals one over a n, we must have a CZ and goes to infinity, that ex close to infinity because one over zero goes to infinity. So here I can replace this with Lim is X approaches infinity. And then how did we rewrite this? Well, so we have Ellen. This is one plus and that's one of Rex. But then on the denominator, that's just one over X. So here, as we take the limit, both numerator and denominator. Well, first of all, the numerator as X goes to infinity, this just goes to Ellen of one, but that zero. And in the denominator we have one over X. But that just goes to zero. So we have a limit of the form zero over zero. This is indeterminate form, so we use Lopez's house rule. So here we take a look. We're going to use Lopez House rules. I abbreviate that appear with the L. H. So we rewrite that limit. But then we take the derivative of the numerator and denominator. So in the numerator we have one over one plus one over X and then by the chain rule. We have this extra negative one over X squared and then on the denominator. He's the power rule there to differentiate one of Rex, and we could cancel the negative one over and square terms. And then we just take the limit of this. And that's just one over one plus zero, which is one. So since this Siri's converges, our Siri's will also convert violin a comparison test because our limit satisfies this inequality. It's a number that's bigger than zero, but less in infinity. No, c was one. In our case, this is what is allowing us to use the Lim comparison test. So also converges Bye, Living Computer Sim, and that's our final answer

So because no submission of the seas convergent so from enough. Yeah, natural enough am and must be smaller than one in actual beacon virgins. And that's why we will have Yeah, I and it would be 1,000,000 square will be smarter than I am. And, uh, and then because of submission that mean is in virgin Didn't buy comparison. There's can conclude that s immersion in square on so convergent

Wonder So much wind is convergent, then Father Limit Under Eye and investing in a Mustang Coaches. Aargh! An instant for for last, enough end for Nash and I and be smaller than going for on. And And if it means that I and square with this morning a hand for our own and last enough, therefore, by the director in person test, the submission of the in square will be convergent.


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