- #1

steelphantom

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I have a few series which I'm having trouble proving whether they converge or diverge. I know the following tests for convergence: comparison test, ratio test, n-th term test, and root test. Here are the series and what I have tried so far:

[tex]\sum[/tex] n -1 / n

[tex]\sum_{n=2}^\infty[/tex] 1 / (n + (-1)

And finally,

[tex]\sum[/tex] n! / n

Thanks for any help!

[tex]\sum[/tex] n -1 / n

^{2}: I'm assuming this series diverges, since it behaves like 1/n, which also diverges. I'm trying to use the comparison test to see if I can find a "smaller" series which also diverges, but coming up blank. I tried the ratio test to no avail, since it gives 1.[tex]\sum_{n=2}^\infty[/tex] 1 / (n + (-1)

^{n})^{2}: I'm really not sure where to begin with this one. The (-1)^{n}is really throwing me off. I'm assuming this converges.And finally,

[tex]\sum[/tex] n! / n

^{n}: I tried the ratio test, canceling out the factorial and getting the ratio of n^{n}/ (n + 1)^{n}. This limit seems to be 1, so the ratio test doesn't really help me here. Any suggestions?Thanks for any help!

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