11302 - Hexadecimal Digits of an Integral

All about problems in Volume 113. If there is a thread about your problem, please use it. If not, create one with its number in the subject.

Moderator: Board moderators

Post Reply
RoBa
New poster
Posts: 8
Joined: Thu Aug 11, 2005 7:57 pm

11302 - Hexadecimal Digits of an Integral

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

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

sclo
Guru
Posts: 519
Joined: Mon Jan 23, 2006 10:45 pm
Location: Vancouver, BC, Canada
Contact:

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
New poster
Posts: 22
Joined: Thu Apr 28, 2005 10:56 pm

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
New poster
Posts: 25
Joined: Mon Sep 03, 2007 10:54 am

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
New poster
Posts: 22
Joined: Thu Apr 28, 2005 10:56 pm

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

Hint: The result is a wellknown constant

baodog
Experienced poster
Posts: 202
Joined: Wed Jul 04, 2007 6:53 am

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

Hint:

Bring integral inside the sum,
Integrate before you sum!

sclo
Guru
Posts: 519
Joined: Mon Jan 23, 2006 10:45 pm
Location: Vancouver, BC, Canada
Contact:

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

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

Robert Gerbicz
Experienced poster
Posts: 196
Joined: Wed May 02, 2007 10:12 pm
Location: Hungary, Pest county, Halasztelek
Contact:

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
New poster
Posts: 25
Joined: Mon Sep 03, 2007 10:54 am

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

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

sclo
Guru
Posts: 519
Joined: Mon Jan 23, 2006 10:45 pm
Location: Vancouver, BC, Canada
Contact:

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
Experienced poster
Posts: 202
Joined: Wed Jul 04, 2007 6:53 am

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.

Post Reply

Return to “Volume 113 (11300-11399)”