Hi all,

In relation to the root test for series convergence, we're given the identity:

$\displaystyle \lim_{n\to\infty}\sqrt[n]{n} = 1$

I got curious about this and came up with at least this being true:

$\displaystyle \lim_{n\to\infty}\sqrt[n]{(an^b +c)^d} = 1$

(Where a,b,c,d are real constants.)

but I'm thinking there are only limited situations for which the following is true:

$\displaystyle \lim_{n\to\infty}\sqrt[n]{f(n)} = 1$

(Where f(n) is any function of n, including, for example, f(n) = n!)

like $\displaystyle f(n) = n^n$ in this situation would diverge.

So, what is the truth table for the above equation? It seems to me that the root test would be easier to conduct with more general forms available, re my first equation versus my second.

Any thoughts appreciated.

Thanks,

Brian