(*) (a_1 + a_2 + . . . )^2 = a_1^2 + a_2^2 + . . . is indeed a fascinating equation to ponder. Assume hereon that a_i -> 0 as i -> oo. Note that since sum (a_i) may not be absolutely convergent (or perhaps *must* not?), the LHS may (must?) depend on the order in which its terms are summed. Maybe the intelligentest order is to do (**) sum {n = 2 to oo} of (sum {i+j=n} of a_i*a_j). The truth of (*) <=> the cross terms (sum {i <>j} a_i*a_j) = 0. Of course, if the cross terms are NOT absolutely convergent, then there are always some (inf'ly many) orderings of them that sum to 0. (Or any other chosen sum.) So unless we make some assumption about the ordering of (a_1 + a_2 + . . .)^2, (*) is not that interesting; I propose (**). What would be really cool to see would be a *geometrical* proof of sum (1/n^2) = pi^2 / 6. That is, some object of volume pi^2 / 6 that can be partitioned into pieces of volume 1/n^2. P.S. Also note that 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - 1/13 . . . . = integral {0 to 1} (1+x^2) / (1+x^4) dx --Dan