Originally Posted by

**ThePerfectHacker** Remember the definition what what is means convergence of a series (we had it on our mid-term).

Consider the sequence of partial sums,

S_1=-1

S_2=-1+1=0

S_3=-1+1-1=-1

And so on,

Thus,

-1,0,-1,0,-1,...

Is the sequence of partial sums.

This sequence has no limit.

There are two ways to show it has no limit.

Either violate the epsilon definition.

But here is an easier way.

Assume that is has a limit (real or infinite).

Then, **all** subsequences must have the same limit (theorem 11.2).

But,

a_even = 0,0,0,0,...

b_odd = -1,-1,-1,...

Are two subsequences that converge to two different values, i.e. -1 and 0 irrespectively.

Thus, there is no limit.