[pdurrant]

Originally Posted by jbjb

Spoiler:

I don't think that explanation of the diverging sum is quite right. For example, if you take the third item in each sequence, the entry from the second sequence is not actually less than or equal to that from the first.

Perhaps a better way would be to group them as:
1/10, 1/20, (1/30 + 1/40), (1/50 + 1/60 + 1/70 + 1/80) ... etc.

In general, the nth group of terms (where n starts at 1 for (1/30+1/40)) will (for all n > 0) be a sum of 2^n fractions, the least of which will be 1/(20*(2^n)). Thus the sum of each group must be at least (2^n)/(20*(2^n)), or 1/20. A sum of an infinite sequence of groups, each of which has a sum of at least 1/20 will itself be infinite.

Cheers,
John

You're entirely correct. I'll change it so it's correct. That's what I get for writing and posting without double-checking. (I usually write the answer to a text file one day, and then re-read and post it the next.)