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Q3 a) Explain why n+I < Jm+i Zdx for each n € N. b) Define a sequence (tn);_1 by tn = (Er_1 }) _ log(n). Show that this sequence is decreasing and that 0 &...

Question

Q3 a) Explain why n+I < Jm+i Zdx for each n € N. b) Define a sequence (tn);_1 by tn = (Er_1 }) _ log(n). Show that this sequence is decreasing and that 0 < tn < 1 for all n. Why does limn-+o tn exist?3

Q3 a) Explain why n+I < Jm+i Zdx for each n € N. b) Define a sequence (tn);_1 by tn = (Er_1 }) _ log(n). Show that this sequence is decreasing and that 0 < tn < 1 for all n. Why does limn-+o tn exist?3



Answers

Show that the sequence defined by
$$a_{1}=2 \quad a_{n+1}=\frac{1}{3-a_{n}}$$
satisfies $0 < a_{n} \leqslant 2$ and is decreasing. Deduce that the sequence is convergent and find its limit.

Welcome back to New Murad. My name is Kevin Chirac. Let's consider the sequence. Ace of an is equal to one divided by three to the end and we're looking for what happens as end goes off to infinity will end goes to infinity three to the end becomes very large and one divided by a very large number will be a very small positive number. So we can say this entire sequence simply converges to zero. I hope this video help.

It's under form three plus course. I'm the high end and go to 123 and so on. So tragic from the anyone included three plus cause I am the pie Cassandra Pye but u minus wonderful GONNA Jr A And you equal to three plus a close eye off the Jew pie for such a prank. But you wonderful mother far a tree include your tree plus a close eye on that three prime way Get echo Judah cassette repackage months wonderful about a Jew And for a good tree bus goes I'm the for pie because I for by could you wonderful gonna fall that every time Doing so was said under I am really got you and we could you do if that And here is what on good to fall if and is even therefore is it implies that the limit of the day and investing in Philip they does not exist here. They found the sequence even be diverted

Welcome back to New Murad. My name's Kevin Chirac. Let's take a look at this. See this sequence here? And we're considering what happens. A Zen grows larger to infinity, but we're gonna rewrite this so it becomes easier to analyze so we could combine. The entire numerator is simply being three end squared plus the rienda and then on the bottom, we could write this is simply too and squared. And by now we can recognize that if we have the same power on the top of the bottom, we're really just looking at the leading to coefficients. That means that the limit as n goes to infinity of Ace of End is going to simply be that ratio 3 to 2, which will be also our limit. Therefore, the sequences convergent. I hope this video helped if it

Giving that a one equals 1 And a. and bloods one equal to this. We first want to show that. Sure that's A. N. Is within the endeavor. So and two for for all in in and which is son. Natural number Sue and equals one is true. If it's true then we have and so you want to be equal to zero is less than a one which is equal to two Less than or greater than two. Assume I assume that for some and in natural number any natural number you have the interval of E. N. Zoo 22 Then this implies that we have so less than E. N. Unless I know equal to two. So you have sarah Greater than -4. Better than or equal to -2. Then you have three great sudden three minutes A. N. Which is greater than or equal to one divided by three. And this is less than okay. So they we have. So this implies that's is in place we have one divided by three less than one divided by three. Mine it's a N Less than or equal to one. which implies that one divided by three is less than A N Plus one. E n plus one less than Oh greater than one. So this implies that zero is less than a N plus one less than or equal to sue. So by mathematica induction. So this way mathematica mathematica induction. In nation we have we get zero to be less of a less than or equal to two is true for all N in natural number. So this implies that the sequence the sequence the sequence is bounded. So if we show if we sure that's if you should that's it is it is manu when a tonic two then then we will. No. The sequence converges. The sequence converges the sequence school pages. So we have a two. Say if we big eight soon. two vehicles 1 divided by three minutes a one This is equal to one, divided by 3 -2, which is equal to one. So this implies that a two is less than a one. So as I zoom, I assume that for some and the natural number A. N. Platz one, A N plus one is less than E. So then it implies that A N plus one is less than E. N. So negative A N plus one will be greater than minus A. So uh three -10. It's greater than three -A. So one divided by three. Mine, it's A N plus one. It's less than one divided by three. My next a N A su we have E N plus two to be less than A N plus one. So then again by mathematica induction, we get that. This implies that a N. Platz one is less than a N for all. And in a natural number. So this implies that the sequence the sequence is decreasing and bounded, which also implies that the sequence is convergence, its conversion. So if we take their limits, so we take the limits as N approaches infinity of a N, it's equal to the limit as N approaches infinity of one divided by three minus A N, which is equal to L. So this implies that L vehicles one divided by three minutes ill. So we have minutes, L squared plus three L to be equals we're which implies that our L is equal to three plus or minus square roots of five divided by two. Sees says three plus square roots of five divided by two is greater dan two. It is, notes the solution. So then it's in place that the limits as N approaches infinity of a N, it's equal to three minus square root of three, Divided by square roots of five, So you have three three minus square roots of five, divided by two as a final answer.


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