Ramanujan Series

[Ollie]

Howdy all. Firstly, apologies if this is in the wrong sub-forum. Now, On with the question!

 

I've recently been reading through Marcus du Sautoys "Music Of The Primes". It's a great book, I thoroughly reccommend it to anyone wanting a book to read. I've been keeping up with what the maths ok so far (precious little that there has been), but one thing's bothered me. There's a chapter about Srinivasa Ramanujan, where it says that in his first letter to Hardy and Littlewood he wrote the line

 

1 + 2 + 3 + .... + n = -1/12

 

where n is infinity. Yes, I know it looks crazy. Apparently it took H&L an evening or two to work out that he really meant:

 

1 + 1/(2^-1) + 1/(3^-1) + ... + 1/(n^-1) = -1/12

 

According to du Sautoy, this makes perfect sense. Now, I'm the first to admit that sums of series have never been my strong point, but this can't be right, can it? Surely the two lines mean the same thing, and the answer is wrong? Apparently this is a solution for zeta(-1), but no matter how I look at it, I always get infinity. What am I missing?

 

On a related (I think) note, would anyone be able to tell me what the evaluation of x^i is? My blind faith that there is a proper answer to that (which is somehow negative) is all that keeps me thinking I understand the Riemann hypothesis (even just a little). If not, then I've got to go back to the drawing board

 

Many thanks from a newbie, Ollie