University of Warwick
Coventry

# Warwick First Year Maths-Past paper questions

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benwellsday
Last minute questionssss
$\sum _n$
What does that mean in analysis? In foundations it says it can denote the permutation group but in the one analysis question where they've used that notation it asks about convergence / divergence which implies it's a series. I assumed it was just shorthand for $\sum _{n=1} ^{\infty}$.
It's not really that important, but I'm bored and have 3 hours to kill...

I thought that just $\sum$ meant the sum to infinity. i think with an N there is means sum to n?
University of Warwick
Coventry
But that still wouldn't make sense to ask about convergence/divergence with a finite sum. It's Part o) of question 1) in 2007, If $\sum_n a_n$ diverges then $\sum_n (-1)^n a_n$ diverges. T/F. Given I understand the notation then it's false with an=1/n. Maybe it was a notation they used more in 2007. If I see it today I'll guess what it means based on what the question is anyway. Actually that's kind of my technique for the whole of analysis...
benwellsday
But that still wouldn't make sense to ask about convergence/divergence with a finite sum. It's Part o) of question 1) in 2007, If $\sum_n a_n$ diverges then $\sum_n (-1)^n a_n$ diverges. T/F. Given I understand the notation then it's false with an=1/n. Maybe it was a notation they used more in 2007. If I see it today I'll guess what it means based on what the question is anyway. Actually that's kind of my technique for the whole of analysis...

well in that context it must mean to infinity as it says diverges but generally i'd have thought it would mean sum to n rather than inifnity.
It means sum over N
is $2^{91}-1$ prime?

at a guess i'd say it is prime, but i just doon't knoww.
benwellsday
Got a problem, 2006 analysis paper, give an example of a bounded divergent sequence such that $\frac{a_{n+1}}{a_n} \to 1$. Old post, but:

Why aren't you guys posting on F38?
DFranklin
Old post, but:

Why aren't you guys posting on F38?

Because we're too cool for F38? Don't know, we could've created this on f38 tbh.
IrrationalNumber
Because we're too cool for F38? Don't know, we could've created this on f38 tbh.

noooo, we're not on F38 because there's so many of us Warickticians here we luvv it.
DFranklin
Old post, but:

Why aren't you guys posting on F38?

i think our version is nicer but cheers anyways.

btw I'm sure kolya posted that last year...
I figured posting here would be better because there's about 5 of us at least and we would all have the same papers and questions. If every university did a similar thing on f38 it'd get clogged up pretty quick. Then again all the difficult looking university work might scare off the people who post the boring threads!

And I still don't get why $\frac {10 + \sin \sqrt{n+1}}{10 + \sin \sqrt{n}} \to 1$ has to happen. I can see that it does happen by looking at the graph of y = sin {x^(1/2)} but is proving it much harder? I think it has something to do with taking approximation of sin(x^(1/2) + epsilon) as you choose epsilon smaller, and knowing that (n+1)^1/2 and n^1/2 eventually become within epsilon of each other.
benwellsday
If every university did a similar thing on f38 it'd get clogged up pretty quick.

Does every university have as many people doing maths there on TSR?
benwellsday
I figured posting here would be better because there's about 5 of us at least and we would all have the same papers and questions. If every university did a similar thing on f38 it'd get clogged up pretty quick. Then again all the difficult looking university work might scare off the people who post the boring threads!

And I still don't get why $\frac {10 + \sin \sqrt{n+1}}{10 + \sin \sqrt{n}} \to 1$ has to happen. I can see that it does happen by looking at the graph of y = sin {x^(1/2)} but is proving it much harder? I think it has something to do with taking approximation of sin(x^(1/2) + epsilon) as you choose epsilon smaller, and knowing that (n+1)^1/2 and n^1/2 eventually become within epsilon of each other.

well we can show that $\frac{\sqrt{n+1}}{\sqrt{n}}$ tends to 1...
IrrationalNumber
Does every university have as many people doing maths there on TSR?

probably not, the only one i can think of that might would be cambridge. but i think out year TSR wise is even bigger than theirs!

plus, F38 is full of ******* posting retarded 'core 1 series' questions.
Totally Tom
probably not, the only one i can think of that might would be cambridge. but i think out year TSR wise is even bigger than theirs!

plus, F38 is full of ******* posting retarded 'core 1 series' questions.

I bet when we look back on these posts in three years time we'll think our questions are retarded.
IrrationalNumber
I bet when we look back on these posts in three years time we'll think our questions are retarded.

possibly, but we don't exactly spam a whole forum up with them.
benwellsday
And I still don't get why $\frac {10 + \sin \sqrt{n+1}}{10 + \sin \sqrt{n}} \to 1$ has to happen. I can see that it does happen by looking at the graph of y = sin {x^(1/2)} but is proving it much harder? I think it has something to do with taking approximation of sin(x^(1/2) + epsilon) as you choose epsilon smaller, and knowing that (n+1)^1/2 and n^1/2 eventually become within epsilon of each other.
|sin(a)-sin(b)|<= |a-b| (if you think it needs proving, prove it by factor formula, or MVT).

And $\sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}} < \frac{1}{2\sqrt{n}}$.

So if $n^2 > 1/\epsilon$, then

$|\sin(\sqrt{n+1})-\sin(\sqrt{n})| \leq \sqrt{n+1}-\sqrt{n} < \frac{1}{2\sqrt{n}} < \epsilon$

Finally, $|\frac {10 + \sin \sqrt{n+1}}{10 + \sin \sqrt{n}} - 1| = |\frac {\sin \sqrt{n+1}-\sin\sqrt{n}}{10 + \sin \sqrt{n}}| < 1/5|\sin(\sqrt{n+1})-\sin(\sqrt{n})|$
Ah that looks better, though not something I would of been able to think of on the spot. Plus I didn't know the |sin(a)-sin(b)| <= |a - b| inequality although in one of the analysis books we're allowed to assume basically the same thing but with cos.
You could also use a sawtooth function $S(x) = \min \{|x-n| : n \in \mathbb{Z}\}$ instead of sin. It's easier to prove it works using only elementary analysis, but a bit fiddly to cover all the details.
DFranklin
Why aren't you guys posting on F38?

Benwellsday
I figured posting here would be better because there's about 5 of us at least and we would all have the same papers and questions. If every university did a similar thing on f38 it'd get clogged up pretty quick.
The idea has been brought up before, but maybe this suggests a use for a (trial) uni-level sub-forum of f38? Then those who can answer won't have to go to an obscure sub-forum of a university to answer the question, and the questions will not be lost in the deluge of A-level posts in f38.
Sounds like a good plan, when's that leader of the TSR math soc thingy going to be elected, I'm sure they could sort something out. And hopefully it could be regulated so that it doesn't get filled with "Which uni should I go to with my grades?" or other pre-uni type questions.
The only downside I could think of is that if everyone is like me they'd go to the uni sub forum first to check threads before helping out with any A-level questions they might know. They can't complain though, they get solutions in the back of their books!