You are Here: Home >< Maths

# The Proof is Trivial! Watch

1. (Original post by Lord of the Flies)

Problem 103* (there is a slick *** solution though)

Noticing the addition formula and regarding e as an arbitrary parameter:

setting e=1 gives C=0 and letting e be eulers constant we get I(e)=pi/2.
I've had a shocking day mathswise, am I close?
2. (Original post by ben-smith)
I've had a shocking day mathswise, am I close?
Can't be worse than me. I've been wrestling with "games" that are ironically not fun at all for the whole day
3. (Original post by Lord of the Flies)
I thought it had been solved! Here we go:

Solution 98

by Stolz-Cesaro.
Hmmm...never heard of that test. It looks pretty cool. However; could you walk me through your justification as to why the original limit exists?

Edit: The one with the a_n and roots on the denominator.
4. (Original post by ukdragon37)
Can't be worse than me. I've been wrestling with "games" that are ironically not fun at all for the whole day
Part III sounds tough

(Original post by Lord of the Flies)
Yep - there is an arctan missing but that is just a typo.
You have me intrigued as to what the slick way is. Complex numbers?
5. (Original post by ben-smith)
Part III sounds tough
Thankfully I only need a pass to move on to PhD. And if my exams go as I planned I should only need 65% in my dissertation to get a distinction (75%) overall (60% is the passmark).
6. (Original post by ukdragon37)
Thankfully I only need a pass to move on to PhD. And if my exams go as I planned I should only need 65% in my dissertation to get a distinction (75%) overall (60% is the passmark).
Are you going to minor in anything cool like, I don't know... maths?
7. (Original post by ben-smith)
Are you going to minor in anything cool like, I don't know... maths?
I was thinking of minor-ing in Law or Music actually Law was my first choice for applying to Cambridge before I changed my mind and went with my second choice, CompSci.
8. (Original post by SParm)
Hmmm...never heard of that test. It looks pretty cool. However; could you walk me through your justification as to why the original limit exists?

Edit: The one with the a_n and roots on the denominator.
hence either or as . A quick contradiction shows that there is no such

(Original post by ben-smith)
You have me intrigued as to what the slick way is. Complex numbers?
You're going to be disappointed - essentially the same as what you did, but set up differently:

Another? You TSR folk are too good!

Problem 105* (technically, if we are rigorous, this is a ***)

(the curly braces don't mean anything in particular - I just thought they looked nice)
9. (Original post by Lord of the Flies)
hence either or as . A quick contradiction shows that there is no such
That bit is fine. I'm just wondering about what happens to the roots at n goes to infinity. Mind it's late and I'm tired but I certainly didn't get 0 for this (Well I didn't then I did by second guessing myself and then I didn't again once I realised I was being a tit.)
10. Problem 106

Prove the formula

Problem 107

Evaluate the series
11. (Original post by SParm)
...
Oh no, you're absolutely correct. I am the one being an idiot. I somehow omitted the roots in my working leaving (n+1) - n = 1. My sincere apologies.

Edit: amended below.
12. (Original post by Lord of the Flies)
Oh no, you're absolutely correct. I am the one being an idiot. I somehow omitted the roots in my working leaving (n+1) - n = 1. My sincere apologies.
It's fine, I'm sure we've all had that one (or few dozen for me) shockers when writing maths down on paper. I'm reluctant to post my solution to this, even if I'm confident it works, simply because this thread is brilliant for pre-undergrad level peeps to have a go at some cool maths. My solving what I think is a lovely Analysis problem with a first year Analysis course under my belt kind of ruins that a bit.

I'm sure Mladenov would be happy to give anyone who wants to do it some hints, as would I.

Edit: Also my solution's laborious to the max. I'm sure one of you guys can come up with a lovely 4 or 5 liner.
13. A lot of big terrifying things flying around this thread
Spoiler:
Show

Problem 108*

Find all pairs of integers (p,q) such that the the roots of the equation, , are integers.

Spoiler:
Show
Mladenov isn't allowed to solve it and I'm sure he knows why

-----------------

Btw, can people make sure they put 'assumed knowledge' ratings on problems please.
14. Solution 98

Square the limit:

then hence by Stolz Cesaro.

Hence

Solution 106

inductively.

Multiply by

Solution 107

15. (Original post by FireGarden)
Problem 106
Prove the formula
...for all non-negative integers n.

Solution 106

When n=0, LHS. RHS. So true for n=0.

Assume true for n=m. This gives...

.

Therefore, if true for n=m, true too for n=m+1. As true for n=0, true for all non-negative integers n by mathematical induction. QED.
Spoiler:
Show
My latex masterpiece
16. (Original post by Lord of the Flies)
Solution 98
\lim_{n\to\infty} \frac{a_n^2}{n}=2+\frac{a_0}{n}+ \sum_{k=0}^{n-1}\frac{1}{a_kn}[/tex]
Awesome solution, Stolz Cesaro seems pretty useful. Where did the squares go though? (even if they were squares the proof would still work so I'm just wondering)
17. (Original post by Jkn)
Awesome solution, Stolz Cesaro seems pretty useful. Where did the squares go though? (even if they were squares the proof would still work so I'm just wondering)
I forgot to type them in! All fixed now, thanks for noticing!
18. (Original post by Lord of the Flies)
I forgot to type them in! All fixed now, thanks for noticing!
Dude, you're way too tired I noticed your little blunder earlier btw! I assumed it was another one of those things that are only obvious to you and mladenov so spent a few minutes trying to convince myself that was divergent! As you can imagine, I was quite puzzled!
19. (Original post by Lord of the Flies)
Solution 106
Oi! You only just edited that in! I better see a "(2)" pretty soon or there's gonna be hell to pay (nice and elegant btw, I clearly missed the point!)
20. (Original post by Jkn)
Dude, you're way too tired
Ha! I think you're right, time for bed!

(Original post by Jkn)
Oi! You only just edited that in! I better see a "(2)" pretty soon or there's gonna be hell to pay (nice and elegant btw, I clearly missed the point!)
I had already solved it but was eager to post a solution to 98, after that blunder. Also, it doesn't matter what order the solutions come in - this thread is not a competition, it's for fun!

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: December 11, 2017
Today on TSR

### What is the latest you've left an assignment

And actually passed?

### Simply having a wonderful Christmas time...

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.