Rearranging The Alternating Series Of Reciprocal Squares

Question

A question has come up on a group project problem for my Calculus II class that I am having a difficult time with.

Here is the question:

"Find a rearrangement of the series $\sum_{i=1}^\infty (-1)^{i+1}/i^2$ that converges to a negative number and provide compelling evidence that the sum is indeed negative.

I'm just not sure how to approach the rearrangement. My partner and I have both tried coming up with a solution, and neither of us can find one.

Riemann's Rearrangement Theorem series says that by changing the order of summation of a conditionally convergent series one can change its value to any real number. Although is there some sort of method that tells you what sort of rearrangement to use? I mean Thomas' Calculus book just basically states the theorem and proves it, and that's it.

Thanks in advance for any help.

Answer 1

Our series is absolutely convergent. All rearrangements give the same sum. So no rearrangement gives a negative sum.

We do not even need to quote the general result. Sooner or later we will have to use the term $1$. From then on, partial sums are certain to be positive.

Remark: Perhaps $\sum_1^\infty (-1)^{i+1}/i$ was intended.