I am investigating the convergence of the following series: 14+1⋅94⋅16+1⋅9⋅254⋅16⋅36+1⋅9⋅25⋅364⋅16⋅36⋅64+⋯
This is similar to the series 14+1⋅34⋅6+1⋅3⋅54⋅6⋅8+⋯,which one can show is convergent by Raabe's test, as follows: an+1an=2n−12n+2=2n+2−32n+2=1−32(n+1).
However, in the series I am looking at, an+1an=(2n−1)2(2n)2=1−4n−14n2, so Raabe's test doesn't work (this calculation also happens to show that the ratio and root tests will not work).
Any ideas for this one? It seems the only thing left is comparison...
Answer
I just read that there is a partial converse to Raabe's test that if an+1an≥1−p/n for p≤1, then the series diverges. So I think that settles that it diverges.
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