I decided to take a look at wikipedia. It reminded me of the integral test.
The sum of the first n terms > integral of 1/x dx from 1 to k+1 = ln(k+1)
Which implies the solution to ln(k+1)=10 is a lower bound.
k = e^10 1 = 22025.46
***edit this is an upper bound***
Below that on the wikipedia entry is the rate of divergence:
The sum of the first n terms = ln(k) + EMC + error
EMC is the EulerMasceroni Constant which is about .5772156649
and the error term is about 1/(2k)
Using the lower bound for the k in the error term gives k=e^(10EMC1/(2*22025.46) = 12366.87
I am pretty sure that the error term is small enough to imply 12366 is the solution to the problem. (12367 would exceed 10)
Edited on November 1, 2011, 10:40 am

Posted by Jer
on 20111101 10:37:54 