I’ve been trying to develop a simple formula, but have hit a rather large problem that I cannot seem to solve. I’ll spare you the details of what I’m trying to do and give you the problem straight off. I apologize for my basic mathematical notation.

How do I work out the sum of n terms in the sequence (1/1)+(1/2)+(1/3)…? For example, how do I find the total of (1/1)+(1/2)+(1/3)+(1/4)+(1/5)+(1/6) without actually adding each term separately? This is, of course, the same as (1^-1)+(2^-1)+(3^-1)… I am aware of formulas for arithmetic and geometric progressions, but this appears to be neither – is it an exponential progression, power law or whatever else? Am I forgetting some fundamental rule?

Whatever steps I’ve taken towards solving this, I always arrive back at the same basic problem. Factorial numbers (x!) seem to turn up often, but that might just be my repetitive thinking. I’ve given up trying to work this out myself.

Expert and maybe not-so-expert mathematicians of the Straight Dope message board, please help me out!