Hey guys, I'm pretty stuck on the following problem. It involves rearrangements of the terms in the alternating harmonic series to produce different sums:
Let s = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...
a) Show that s/2 = 1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + 1/5 - 1/10 - 1/12 + ...
b) Show that 3s/2 = 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ...
I think I've got part (a) so far. If you look at each group of three terms, pairing the first two terms in that group together and computing the difference seems to do the trick.
(1 - 1/2) - 1/4 + (1/3 - 1/6) - 1/8 + (1/5 - 1/10) - 1/12 + ...
= 1/2 - 1/4 + 1/6 - 1/8 + 1/10 - 1/12 + ...
= 1/2 (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...)
= s/2
I'm stuck on part (b) though. I've tried many different groupings, and also tried using the result from part (a) and adding it to the given information to try and get the result, but I just can't seem to get it. Any tips or advice on this part would be greatly appreciated. Thanks!
EDIT: Finally figured it out. I subtracted the sum for s/2 from the sum for 3s/2, term-by-term, and showed that those terms were in the right order to produce the sum s. I thought I tried this earlier but I guess I just got hung up and confused on the grouping aspect of it.