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Prove the following estimates when n - OO0 _In n o(n). (b) In n o(n4/2) . In n o(nQ.0001 ) ....

Question

Prove the following estimates when n - OO0 _In n o(n). (b) In n o(n4/2) . In n o(nQ.0001 ) .

Prove the following estimates when n - OO0 _ In n o(n). (b) In n o(n4/2) . In n o(nQ.0001 ) .



Answers

Prove that $n<2^{n}$ for all natural numbers $n$

Okay, so here um we let absalon be greater than zero and then we have that since a Saban approaches lambda and be suburban approaches M. Therefore there must exist a positive integer, M one and M two. Such that the well we have the absolute value of a sub n -L is going to be greater than 1/2 um epsilon. And then we have that the absolute value of a sub N -L is going to be greater than one half. Absalon for all and greater than or equal to M one, and we have the absolute value of B sub n minus M is going to be less than um epsilon over to for all and greater than or equal to M two. So now let em Um be the max let em be equal to the maximum of M one and M two. Then we have that the absolute value of a sub n minus L is going to be greater than one half absalon. And we have that the absolute value of B sub n minus M is going to be less than epsilon over to for all and greater than or equal to m. Okay. And then we have that the absolute value of a sub n plus B sub n minus L plus M is going to be equal to the absolute value of a sub n minus L. Plus the absolute value of B sub n um minus M. Which is going to be less than or equal to the absolute value of a sub n minus L. Plus the absolute value of B sub n minus M which is going to be equal to epsilon over to plus epsilon over two, which is equal to epsilon. So therefore that implies here that S M N um plus this Evan is going to approach our plus M and therefore the proof is complete.

In this problem of mathematical induction we have to prove. Given a statement using principle of mathematical induction for all and belongs to natural number. First we consider given a statement as pr friend which is Who Ain't last seven less than N. Plus three whole leesburg. First we prove 40 of one Putting anywhere to one. So we have here nine less than this will be 40 square 16. So it is true for P. Of one. Now we consider pr escape. So it will be who gave Last seven less than K-us three. Police work. Now we need to prove for p. of gay plus one. So it will be oh K plus one. Last seven. So we write it mm. Okay plus two plus seven. Again we write eight is okay plus seven plus two. Here we can say we can see P. F. K. Two. K plus seven is less than K plus three. Holy square. So figure right here, less than K plus three. Holy square plus two. After expanding this, we have value equal to okay square. Last nine plus six games plus two. So this will be a case where plus six Cape plus 11 from the origins have given a statement. We have a value equal to mm hmm K plus one Plus three. So it will be 12 K-us four. Police work. So it will be greater than this expanding this. We can observe. So it will be K squared plus eight. K plus 16 K. Is alienation number. So it is greater than this expression. So we write it here. K square plus six K plus 11. Greater than K square plus eight K. Mhm. Last 16 again. We can write it S. K. Plus for Holy Square. And this is less than this is greater than this value. So we can write it here. Okay Plus two plus 7 again, simplified expression two K plus one plus seven is less than K plus four. We can write eight S. K plus one last three. Holy square. So it is also True. four K Plus one. So we can write it. Mhm. Mhm. Given a statement, look at the people, it's true for all and belongs to natural number. And this will be our final answer yeah.

Hey guys, In this problem, the textbook asks us to prove that n factorial is equal toe end times and minus one factorial. Now, in order to do this, we're going to be using the definition of the factorial, so let's get started. So first, let's start with the definition of an factorial. Well, n factorial is nothing but and times and minus one times and minus two times love, love, like all the terms in between, all the way to to times one. Now, another key note that we're gonna make on the side is we're gonna note the side the definition of n minus one factorial well and minus one factorial is the same thing as an minus. Uh, whoops and minus one. And mine is too Times all the way to two times one. Now take a look here. The definition of n minus one factorial is right here. So all the terms that comprise of n minus one factorial are included in and factorial so we could replace all of these terms with and minus one factorial. And in order to do that, we just substituted. So we say that this is equivalent end times and minus one factorial. And voila! We have the proof of the problem. We basically just showed that an factorial is equal to end times and minus one factorial Q e d or the end of the proof. Thanks for listening, guys. And I hope this helped, uh, show you guys how to understand this proof of concept that factorial is equal to end times and mine.

In this problem of mathematical induction we have approved given a statement using principle of mathematical induction for all end belongs to natural number first. Well given a statement sp orphan And we have given a statement one plus two plus three plus up to plus and Less than one upon 8 who and plus one holy square first we prove forward. We of one year of one is basic statement and one is the smallest national number. So we'll take alleges, you know putting anywhere to one. We have very equal to one. Now we take our it is putting an equal to one here. So we have valuable to two plus 13 square upon eggs. So it will make well too nine Upon a. This is required to greater than one so we can say. And the case they satisfy dissatisfied are too for the we are of one for the given statement. Now we consider P. Of gay or to get them. So we write the given a statement one plus two plus three. Plus up to Get them. And this is less than one upon 8 Who gave plus one hole is work. Now we need the proof. We are of K plus one up. Ok plus one time. So we write LHs considering P R. K. Yeah to get them plus Hey Plus 1 to Talk. Now we could be of gay which is one upon it. To K plus one hole is work. So we write it. Mhm. One upon a. Who? K-us one. Holy square plus K plus one. Now we simplify the expression. Taking L. C. M. So L. C. M will be eight and expand these two K plus one. We have value four case for plus one plus 40 and multiplying this aid in two K plus one. We have eight K plus eight. Now we simplify the numerator. Mhm So it will be 4K square plus 12 G plus nine. I'm on a mm. We can write it as Okay Plus three. Hold you as to gays were four cases were plus to multiply by okay Multiply by three plus threes work. And this is also where two cases where so we can write it. Mhm. Cook a plus three. Holy square upon a. Mhm. Yeah. Again arranging this term. So variety too. K plus one plus one. Holy square upon a. Now we take our bridges are just putting N equal to K plus one. So I saw you are riches of the given statement one upon it too. And plus one hole is square. You should be greater than a plus one. Mhm. Two N plus one. Always were upon it putting in equal to K plus one. So we have to. Okay plus one. Mhm Plus one Whole is good upon eight. These two statements alleges our ages are equal so we can say mhm. Mhm. Given a statement. Yeah, yeah, it's true for all and belongs to national number. And this will be our final answer delivery.


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