Dirichlet test for convergence

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SUMMARY

The discussion focuses on applying the Dirichlet test for convergence to the series 1 - 1/2 - 1/3 + 1/4 + 1/5 - 1/6 - 1/7 + ... The participants emphasize the importance of correctly identifying the components of the series, suggesting that it should be factored appropriately rather than split into separate series. The key to applying the Dirichlet test lies in determining a bounded sequence and a decreasing sequence that converges to zero, specifically using the sequence a_n = {1, 1/2, 1/3, 1/4, ...} as a basis for multiplication to reconstruct the original series.

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Homework Statement


use dirichlet test to determine if the series converges 1-1/2-1/3+1/4+1/5-1/6-1/7+...



Homework Equations





The Attempt at a Solution


I have broken up the series into two different series the first series I have is
1+1/4+1/8+1/12+... and the second series I have is -1/2-1/3+1/5-1/6-1/7+...

I'm not even sure if these are the two correct series to break it up into, but I know I need to determine if one is bounded and the other is decreasing and limit convergent to 0. Any help is great.
 
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You can't split up the series like that unless you already know it's absolutely convergent. And I really don't see why you would want to. To apply the dirichlet test you want to factor the series. Let a_n={1,1/2,1/3,1/4...}. What series would you MULTIPLY that by to get the original series?
 

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