I'm looking into this for a computer program actually, so I don't need it in mathematical notion - just something I can put into an algorithm :)
Also if its easier with a finite limit on the series then that's ok. I just can't find or work out an inverse of it :S
The logarithmic integral function, which is what you get by integrating 1/ln(x), is closely linked to prime numbers. It approximates the number of primes smaller than x. Heres an infinite series which describes the function:
li(x)=\gamma+ln(ln(x))+\sum^{\infty}_{n=1}\frac{ln(x)^n}{n*n!}...