It diverges???

[Guv]

Did you know that the following series diverges? Id est: It grows without an upper bound.

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 . . . 1/n . . .

It eventually gets big beyond belief.

[parksie]

Nope...but then we get into the realms of infinity which REALLY f**ks my brain over...so I try not to think about it too much.

I just thought it gets closer to infinity as you sum to infinity?

[jmcswain]

You must be thinking of

1/2 + 1/4 + 1/8 + 1/16 + 1/32...

which approaches 1. The arithmetic variation

1/2 + 1/3 + 1/4 + 1/5  + 1/6..

is not only bigger at each stage, but the margin by which it is bigger grows as well, thus resulting in what you originally stated.

[Guv]

While I have known for a long time that this series diverges, I never realized how slowly it grows. I just used my mathCad Software to do some calculations.

Here are the results.

07.485 at 1000 terms.
09.788 at 10000 terms.
12.090 at 100000.
14.393 at one million terms.

It is not growing without bound very fast. If I did the calculations correctly, a person might think that it does not diverge.

There is a proof that it grows without bound, and I always believe valid proofs.

[Sam Finch]

The proof is contained here as well as a proof that the sum of the reciprocals of the prime numbers diverges.