Show that the series sum(-1)^(n-1) b_n, where b_n = 1/2 if n is odd and b_n = 1/n^2 if n is even, is divergent.

Answer 1

we know that ∑1/n^2 converges (so ∑1/(2n)^2 will as well), so this two-fold series will converge if the 1st part converges.

But that is 1/2-1/2+1/2-1/2 ... which can be either 1/2 or 0.
So, the whole shebang diverges.