yup thats the intended solution, im not really familiar with taylor series yet, but maybe for a person who knows taylor series would be able to see it right away
There closest I was able to do on the fly was to show that the ::: pairwise sum has an integral of -ln(2)/2 ::: but that was pretty hand wavy. I'll have to check how it's actually done.
minus one plus a half minus a third plus a quarter... this converges to a value somewhere between 0 and -1, but I dont know how to get more precision without just running this out by hand a bunch more.