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1. Problem 301**

Prove, given

2. (Original post by FireGarden)
Problem 301**

Prove, given

Is there a nicer way than the rather horrible:
Spoiler:
Show
tan both sides, and addition formula
3. (Original post by Smaug123)
Is there a nicer way than the rather horrible:
Spoiler:
Show
tan both sides, and addition formula
Yes. The formula arises naturally, if you consider the right objects/behaviour. I wrote the question with the nice answer in mind
4. (Original post by Smaug123)
Is there a nicer way than the rather horrible:
Spoiler:
Show
tan both sides, and addition formula
That's not that horrible though, it's only 3-4 lines of working.
5. (Original post by james22)
That's not that horrible though, it's only 3-4 lines of working.
It's not pretty, though - it's really mechanical
6. (Original post by FireGarden)
Problem 301**

Prove, given

Solution 301

Here's a nice one from the Putnam.

Problem 302**

Let For and let .

Evaluate:

.
.
7. (Original post by MW24595)
Solution 301

Here's a nice one from the Putnam.

Problem 302**

Let For and let .

Evaluate:

.
.
Solution 302:
Consider:

Suppose then that: is of the form:

For some constant
As it is true for and Suppose that it is also true for

Let:

Observing that:

Where:

This is of the same format as was proposed:

Now using the fact that:

Note that:
and that

So:

Subbing in:

So:

Note that:
8. (Original post by MW24595)
Problem 302**

Let For and let .

Evaluate:

.
If only they had Putnam in the UK..

Solution 302 (2)

Calculating the first few terms we conjecture that .

This is true for n=1 as . Integrating both sides, using integration by parts, formalizing by noting that and factorising on the left hand side shows that, under this hypothesis, we have that:

which completes the induction.

This then gives

where denotes the nth harmonic number.

Using the asymptotic relation , which follows directly from the definition of the Euler Mascheroni constant, we get:

as the logarithmic function is obviously divergent under this limit.

Edit: Dammit, too slow!
9. (Original post by MW24595)
I just came across this. There's an off reference in the section titled "Integration Problems." Read the last paragraph.

http://en.wikipedia.org/wiki/Gamma_function
Oh cheers, must've missed that! (That page is one of my 'most-visited' )
(Original post by MW24595)
Solution 300
Nice solution broseph! Here is your prize!
10. (Original post by Jkn)
Oh cheers, must've missed that! (That page is one of my 'most-visited' )

Nice solution broseph! Here is your prize!
Roflol. I'm surprised you did.
Interesting that the reference was almost bang-on though.

Lol, thanks. Ha

Found anything else of interest lately?
11. (Original post by FireGarden)
Problem 301**

Prove, given

I just realized.

Alternate Solution 301

Let and .

Then, , and, .

.

But, .

Taking logarithms, we have:

.
12. Just posted this in one of the regular threads.. thought you guys might like it

Problem 303**

Give a justification for the claim
13. (Original post by MW24595)
I just realized.

Alternate Solution 301

Let and .

Then, , and, .

.

But, .

Taking logarithms, we have:

.
This is the solution I had in mind. Well done sir!
14. (Original post by FireGarden)
Just posted this in one of the regular threads.. thought you guys might like it

Problem 303**

Give a justification for the claim
This question doesn't really make sense, do you mean we need to find a function, f(x), such that as x->infinity f(x)->1 but f(x)^x->c>2?
15. (Original post by james22)
This question doesn't really make sense, do you mean we need to find a function, f(x), such that as x->infinity f(x)->1 but f(x)^x->c>2?
You could do it that way if you like. I thought the question would make sense, since I'm asking about an indeterminate form, and thus it can take many values - the question is to find some way to approach the IF so that its value is larger than two, as you correctly interpreted. Though your approach is not the only one!
16. (Original post by FireGarden)
Just posted this in one of the regular threads.. thought you guys might like it

Problem 303**

Give a justification for the claim
Solution 303:
Note that:

Now let:

Applying L'Hopital's rule:

17. ...Well that sure does it, but what a meal!

Spoiler:
Show

Mine:

18. (Original post by FireGarden)
...Well that sure does it, but what a meal!

Spoiler:
Show

Mine:

Yeah, I came across the limit problem above earlier, so pretty much just copied and pasted it
I considered;
but it didn't work so I gave up.
19. (Original post by Jkn)
Problem 161**/***

Let k be an integer greater than 1. Suppose and for

Evaluate
I hadn't actually tried this problem when I posted it (as it was a 'problem 6' from the Putnam exam and I was hunting around for things that were going to challenge people), but I've given it a go and just managed to solve it! Really enjoyable so I hope people take a crack at it!
20. (Original post by FireGarden)
Just posted this in one of the regular threads.. thought you guys might like it

Problem 303**

Give a justification for the claim
(Original post by joostan)
Solution 303
(Original post by FireGarden)
...Well that sure does it, but what a meal!

Spoiler:
Show

Mine:

Spoiler:
Show
Not to be a ****, but surely you can both see that all of this is utterly wrong?

If you are intending it as a 'joke problem' of the sort where the object is to spot the error (e.g. prove that 1=2), then please make sure that this is clear in the statement of the problem.

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Updated: December 11, 2017
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