How do i find out if these converge/diverge:
I dont know what K! is and i dont know what to do with the square root. It stumps me and I dont know what theorems to use. The integral test? Bounded sum test? p-series test?
The actual full statement is based on limsup and liminf. The test says given . Consider , the beauty here is that this limit exists (or ) not matter what, the reciprocal will be the radius of convergence. The ratio test is not superior, for one thing we require that the terms be non-zero to use this test! While in root test this is not a necessity! Furthermore, the ratio test is based on root test, because of the fact that: . Thus, if then and we have convergence. If then and we have divergence. All because of the root test. And finally we can give examples where there ratio test fails and root test succedes.