^{1}=1, i

^{2}=-1, i

^{3}=-i i

^{4}=1, you need to separate the series into four parts. S

_{n}as the sum of S

_{4n}.S

_{4n+1},S

_{4n+2}, S

_{4n+3. }Then you can use any test to clarify congergence/divergence of each parts. It seems to me that the series converges.

Bobosharif S.

tutor

Glad to hear that. Good luck!

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02/11/18

Marina P.

Thank you. Can I just ask, lets say I have a sum that diverges absolutely and I'm not sure that it will converge conditionally. Then can I do the separation like that on my sum? We studied that we can only do it if we are sure that our sum converges.

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02/11/18

Bobosharif S.

tutor

Yes, you can that. If you are sure that the series diverges absolutely, it still might converge, as you said, conditionally. Moreover, once you separated a series into several "partial" series, some of them might converge some might not. If we know that the series converges, then all the "partial"'s should converge, otherwise there should be at least one of them divergent.

What you mentioned about convergence, I guess it should slighly different: if series converges, then all partial sums converges as well.

If you like show me that series.

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02/11/18

Marina P.

I get it now, I found it actually written in the book as you said. No need to send, I've solved it, and thank you, you've been so helpful.

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02/11/18

Marina P.

02/11/18