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# The Proof is Trivial! watch

1. Anyone for some analytic number theory?

Problem 525***

Let be a real number. Prove that that there exists a constant such that

2. (Original post by Indeterminate)
x
Wot. I'll see if I can wrap my head around that later, looks fun!
3. Problem 526*

If is an increasing function then show that .

Hence approximate for large .
4. Problem 527**

Let be a large real number. Explain briefly why there must be exactly one root of the equation with . Why is a reasonable approximation to ?

By writing find an approximation on and hence improve on as an approximation to .
5. Solution 526
Spoiler:
Show

This is so much easier to do if I could sketch the curve grr. Consider a rectangle of height f(n) and width 1. This has area of f(n). The top-line of this rectangle lies above-or-on-level with the curve y = f(x) for since f is increasing and hence

Similarly the curve y = f(x) likes above-or-on-level with the line y = f(n) for and hence

Combining these two inequalities gives the desired result, *.

Now let . Therefore for all x > 0 (the domain of g). Hence g is an increasing function so by *:

Summing this inequality from n = 2 to n = N:

. This provides an interval estimate for

Warning I did this entirely typed out in LaTeX so it probably contains some errors so if you see anything please point it out.
6. (Original post by 16Characters....)
Solution 526
Very nice, I just realised that I forgot to prove that was strictly increasing in my proof. This was a Pembroke interview question, if you're interested!
7. (Original post by Zacken)
Very nice, I just realised that I forgot to prove that was strictly increasing in my proof. This was a Pembroke interview question, if you're interested!
Oh right, nice. I have not been able to find any interview questions for my college so I'll just stick to those Trinity ones for now. There is a similar STEP question to this one in the Advanced Problems in Mathematics booklet (Question 27) which finds estimates to factorials by comparing an exact integral to its Trapezium Rule approximation.
8. (Original post by 16Characters....)
Oh right, nice. I have not been able to find any interview questions for my college so I'll just stick to those Trinity ones for now. There is a similar STEP question to this one in the Advanced Problems in Mathematics booklet (Question 27) which finds estimates to factorials by comparing an exact integral to its Trapezium Rule approximation.
St. John's, right? I haven't been able to find any for King's either, so I'm making do with what I find. (although with what I've seen with your mathematical ability, you don't need any prep!)

Yes, I think I recall that vaguely, although I encountered it in a past paper instead of the booklet!
9. (Original post by Zacken)
St. John's, right? I haven't been able to find any for King's either, so I'm making do with what I find. (although with what I've seen with your mathematical ability, you don't need any prep!)

Yes, I think I recall that vaguely, although I encountered it in a past paper instead of the booklet!
Yep St John's. And thank you, the same can certainly be said about you. That problem from the Trinity questions, the one you posted here as 527. I don't know where you'd even begin with that one.
10. (Original post by 16Characters....)
Yep St John's. And thank you, the same can certainly be said about you. That problem from the Trinity questions, the one you posted here as 527. I don't know where you'd even begin with that one.
I gave it a go on the trinity solutions page, first post on page 2 - but couldn't get further than that.
11. (Original post by Zacken)
I gave it a go on the trinity solutions page, first post on page 2 - but couldn't get further than that.
Seen it. I'll leave it to DF to was his way through that one haha.
12. Solution to Problem 527:
Spoiler:
Show
Observe first that is a continuous function.
Now, for , since is very large, we may assume this holds.
Also:

For
Hence by the Intermediate Value Theorem, such that
Suppose that such a solution was not unique, then let two such distinct solutions be and .
Then:

w.l.o.g, let

But:
A contradiction, meaning that .

Now if

As for reasonable sized which occur when is large, we have .

Letting we see that:

For large we may write .
.

13. (Original post by joostan)
Solution to Problem 527:
Very, very nice!

I urge you to just copy-paste this into this.
14. Problem 528:

Which is greater for large , or , justify your answer.
15. Problem 529

Evaluate:

16. Solution to 529:
Observe that:

17. Solution to 528

Clearly .
From this we can deduce

Taking the logarithm again

Now taking the logarithm of twice

So for
18. (Original post by Hauss)
Solution to 528
x
I did this, except I used instead. You might want to add a quick note about being a strictly increasing (continuous bijective) function so that your assertions ring true.
19. Problem 530: */**
Let be arbitrary constants.
Find all possible values of:
20. (Original post by joostan)
Problem 530: */**
Let be arbitrary constants.
Find all possible values of:
Is the denominator of b/etc correct?

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