SOLUTION: Is this infinite Series,convergent or divergent? 1 - 1 + 1 - 1 + 1 ...(- 1)^r

SOLUTION: Is this infinite Series,convergent or divergent? 1 - 1 + 1 - 1 + 1 ...(- 1)^r - 1

Question 1009855: Is this infinite Series,convergent or divergent?
1 - 1 + 1 - 1 + 1 ...(- 1)^r - 1
Answer by ikleyn(52793) (Show Source): You can put this solution on YOUR website!
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It is not converged.

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Comment from student: Hi you say (- 1)^r - 1 is divergent yet (- 1)^r - 1 x b can be proved convergent if b > 0
This can be easily seen if b = 1/n...now if b = 1 the above doesn't change yet you claim it is still convergent..can you explain why?
Best wishes,thx,Colin
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My response.

1) Yes, the series {  } with the constant b, |b| < 1, is converged. It is geormetric progression,  by the way.

2) Yes, the series  is not converged. It is diverged.

These two series are different. They are not the same. Do not miss them. 
Different series have different properties. 

3) Read these two links

https://en.wikipedia.org/wiki/Convergent_series

and

http://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx
(example 3).

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