Welcome to NRICH.

 
A Logarithmic Series
By Niranjan Srinivas on Monday, July 02, 2001 - 09:22 am:

Sum to n terms and to infinity when |x| < 1 :
log(1+x)+log(1+x2) + log(1+ x4) + log(1+x8)...

By Geoff Milward on Monday, July 02, 2001 - 12:59 pm:

log(1+x) can be written as a power series.
log(1+x) = x - 1/2 x2 + 1/3 x - 1/4 x4 + ...
Try writing the power series to x8 for each of the terms in your list above.
I.e. log(1+x)+log(1+x2) + log(1+ x4) + log(1+x8)
and add the coeffiecients of x to form a sinle power series. Do you recognise it?

Have a go and message again if you get stuck

Geoff

By Michael Doré on Monday, July 02, 2001 - 03:08 pm:

Alternatively, you can prove directly that:

log(1 + x) + log(1 + x2) + ... + log(1 + x2n-1) = log((1 - x2n)/(1 - x))

for each natural n.

The result is trivial for n = 1. And if it holds for n then upon adding log(1 + x2n) to each side you get:

log(1 + x) + log(1 + x2) + ... + log(1 + x2n) = log((1 - x2n)/(1 - x)) + log(1 + x2n) = log((1 - x2n)(1 + x2n)/(1 - x)) = log((1 - x2n+1)/(1 - x))

Hence result by induction. I'm sure you can figure out the result for the infinite product from here.

By Levu Boaz Antfalo on Wednesday, September 19, 2001 - 12:01 pm:

Dear Sir,
I don't know whether it is appropriate to ask you this...
I would like a brief history on the topic of logarithms; founder and origin, basically the background information.

Thank-you,
Levu

By Arun Iyer on Thursday, September 20, 2001 - 07:03 pm:

Levu,
You can find the things you wanted at this site.
http://forum.swarthmore.edu/dr.math/problems/temple.7.12.96.html

Hope this helps.

love arun