No actual problem, thinking about the telescoping series theorem and Grandi's series
For reference Grandi's series S = 1 - 1 + 1 - 1...
The telescoping series theorem in my book states that a telescoping series of the form (b1 - b2) + ... + (bn - bn+1) + ... converges IFF limn->inf bn exists.
S can be written (1 - 1) + (1 - 1) + ...
so following the template of the telescoping series theorem, bn = 1.
The Attempt at a Solution
Since limn->inf bn = 1 then by the telescoping series theorem S converges.
If I think about S as a sequence of partial sums, specifically
S1 = 1
S2 = 1 - 1 = 0
S3 = 1 - 1 + 1 = 1
then this sequence diverges and thus S diverges. I don't know how to resolve this contradiction.
I'm really really sorry if this is a common question. I googled this, did a search of this forum, and read about Grandi's series on wikipedia. I haven't been able to find an explanation that I can absorb that satisfies me.