11302 - Hexadecimal Digits of an Integral

RoBa · Post by **RoBa** » Mon Oct 01, 2007 6:47 am

I think it's very funny, but the formula is rather complicated. Any hints?

sclo · Post by **sclo** » Mon Oct 01, 2007 8:30 am

I computed the second sum to be ln(1+x^2), but haven't figured out a closed form for the first product yet. I was never good with working with products.

Tamagodzi · Post by **Tamagodzi** » Mon Oct 01, 2007 3:20 pm

The hexadecimal expansion of 1/11 isnt 0.0F0F.. that is the hexadecimal expansion of 1/17 ...

goodwill · Post by **goodwill** » Mon Oct 01, 2007 3:38 pm

The first product is ln2 but i still don't know what is the value of ln(1+x^2)/(x+1)dx

Tamagodzi · Post by **Tamagodzi** » Mon Oct 01, 2007 4:21 pm

Hint: The result is a wellknown constant

baodog · Post by **baodog** » Mon Oct 01, 2007 7:09 pm

Hint:

Bring integral inside the sum,
Integrate before you sum!

sclo · Post by **sclo** » Mon Oct 01, 2007 7:32 pm

how do you show that the first product is ln(2)?

Robert Gerbicz · Post by **Robert Gerbicz** » Mon Oct 01, 2007 11:52 pm

Probably the fastest way to figure out the constant is to use numeric integration using only say the first 200 terms in the product and in the sum and then from the answer you will know the constant.

You can use PARI-Gp for numeric integration, it's free.
I've used this way in the contest.

goodwill · Post by **goodwill** » Tue Oct 02, 2007 7:42 am

AC Finally. Thank you all. Just "guess" the constant.

sclo · Post by **sclo** » Tue Oct 02, 2007 8:02 am

It is quite difficult to derive a closed form without use of a computer.

baodog · Post by **baodog** » Sat Oct 06, 2007 11:59 am

Tamagodzi wrote:The hexadecimal expansion of 1/11 isnt 0.0F0F.. that is the hexadecimal expansion of 1/17 ...

Thanks. I have sent in correction.
It should have said 1/11 in hex.. 11 in hex is 17 in decimal.
But I can see how that can be confusing, since 36 and 48 in the
problem are in dec, not hex.