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

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Question 1009855: Is this infinite Series,convergent or divergent?
1 - 1 + 1 - 1 + 1 ...(- 1)^r - 1
Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
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 { %28-1%29%5E%28r-1%29%2Ab%5Er } with the constant b, |b| < 1, is converged. It is geormetric progression,  by the way.

2) Yes, the series %28-1%29%5E%28r-1%29 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).