In recent posts, we’ve seen the curious phenomenon that the commutative and associative laws do not apply to a conditionally convergent series or infinite product. Here’s another classic example of this fact that’s attributed to Cauchy.
In yesterday’s post, I showed that
This can be (sort of) confirmed using commonly used technology — in particular, Microsoft Excel. In the spreadsheet below, I typed:
- 1 in cell A1
- =POWER(-1,A1-1)/A1 in cell B1
- =B1 in cell C1
- =A1+1 in cell A2
- =POWER(-1,A2-1)/A2 in cell B2
- =C1+B2 in cell C2
- Then I used the FILL DOWN command to fill in the remaining rows. Using these commands cell C10 shows the sum of all the entries in cells B1 through B10, so that
Filling down to additional rows demonstrates that the sum converges to 
Here’s the result after 2000 terms:
20,000 terms:
And finally, 200,000 terms. (It takes a few minutes for Microsoft Excel to scroll this far.)
We see that, as expected, the partial sums are converging to
Mean Green Math 
I suggest adding two adjacent terms at a time. This will show the convergence faster, and get rid of most of the oscillation.
I don’t disagree; however, my primary point was to illustrate how slowly the unaltered series converges, as opposed to techniques for accelerating the convergence.
That figures!