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11302 - Hexadecimal Digits of an Integral

Posted: Mon Oct 01, 2007 6:47 am
by RoBa
I think it's very funny, but the formula is rather complicated. Any hints?

Posted: Mon Oct 01, 2007 8:30 am
by sclo
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.

Posted: Mon Oct 01, 2007 3:20 pm
by Tamagodzi
The hexadecimal expansion of 1/11 isnt 0.0F0F.. that is the hexadecimal expansion of 1/17 ...

Posted: Mon Oct 01, 2007 3:38 pm
by goodwill
The first product is ln2 but i still don't know what is the value of ln(1+x^2)/(x+1)dx

Posted: Mon Oct 01, 2007 4:21 pm
by Tamagodzi
Hint: The result is a wellknown constant

Posted: Mon Oct 01, 2007 7:09 pm
by baodog
Hint:

Bring integral inside the sum,
Integrate before you sum!

Posted: Mon Oct 01, 2007 7:32 pm
by sclo
how do you show that the first product is ln(2)?

Posted: Mon Oct 01, 2007 11:52 pm
by Robert Gerbicz
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.

Posted: Tue Oct 02, 2007 7:42 am
by goodwill
AC Finally. Thank you all. Just "guess" the constant.

Posted: Tue Oct 02, 2007 8:02 am
by sclo
It is quite difficult to derive a closed form without use of a computer.

Posted: Sat Oct 06, 2007 11:59 am
by baodog
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.