Without going into a detailed analysis, I am sure that this series diverges for all (fixed) values of x.
and the differences between values of for consecutive integer values of don't form any obvious pattern.
However (using a spreadsheet to do the number-crunching) I have found that for all the values of I've tried so far (which is up to ), . Which is what made me think it might be worth looking at powers of or the binary representation of .
Further thoughts: as increases through consecutive integer values:
- the value of the first term, increases by every second value of ; i.e. when
- the value of the second term, increases by every 4th value of ; i.e. when
- the value of the third term, increases by every 6th value of ; i.e. when
- ... and so on.