The second one has a fascinating solution!
We should know the definition of
Ratio test: .
. . . . . . . . .
Divide numerator and denominator by
Since the ratio is , which is less than 1, the series converges.
Hi I am supposed to determine if the following series converges or diverges. The catch is that we are only allowed to use the comparison test, ratio test, and root test.
First one is
and 2nd one is
any hints or the full blown solution would be great.
here's what i came up with
using root test
i guess here is where i get stuck
i have to go to class right to turn this in, but i should that it diverges by comparing it to which can easily shown that it diverges.
You have (every term but the extreme ones cancel : "telescopic sum"). Since when , the series diverges (cf. the very definition of convergence). This is WAY more elementary than any kind of test : just using the definition of convergence.
(if by any chance the square brackets [ ] mean "integer part", then every term in the sum is zero so the series converges)