- #1

marenubium

- 9

- 0

## Homework Statement

No actual problem, thinking about the telescoping series theorem and Grandi's series

For reference Grandi's series S = 1 - 1 + 1 - 1...

## Homework Equations

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The telescoping series theorem in my book states that a telescoping series of the form (b

_{1}- b

_{2}) + ... + (b

_{n}- b

_{n+1}) + ... converges IFF lim

_{n->inf}b

_{n}exists.

S can be written (1 - 1) + (1 - 1) + ...

so following the template of the telescoping series theorem, b

_{n}= 1.

## The Attempt at a Solution

Since lim

_{n->inf}b

_{n}= 1 then by the telescoping series theorem S converges.

If I think about S as a sequence of partial sums, specifically

S

_{1}= 1

S

_{2}= 1 - 1 = 0

S

_{3}= 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.