How can I show that the series [(-1)^n]/n^(1/3) converges?

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- May 19th 2008, 03:44 PMjules027Series convergence
How can I show that the series [(-1)^n]/n^(1/3) converges?

- May 19th 2008, 03:57 PMMathstud28

By the alternating series test since

so for an arbitrary value we have showed that for every value greater than that value that

Now we must show that

So we have also established that

So this series conditionally converges

but it does not absolutely converge since

diverges by the integral test

since it meets all its requirements and

- May 19th 2008, 04:42 PMjules027
That makes sense. This was from a bigger problem: "Give an example of two convergent series

**an**and**bn**such that the series(**an*****bn**) diverges. So would**an**being [(-1)^n]/n^(1/3) and**bn**being [(-1)^n]/n^(1/2) work for this problem? - May 19th 2008, 04:54 PMMathstud28