I was practicing up on my convergence tests using one of those archaic Dover books that are chalk full of good info but are near unreadable and formal/difficult to boot. And I was doing the "Very difficult" convergence excersises and I came up with two questions

The first is that this was supposed to be one of the hardest problems in the book...so I must be doing something wrong..Don't get me wrong..I have condfidence in myself and my answer but this can't be this easy if this is one of the hardest in the book

"Find the convergence for $\displaystyle \sum_{n=0}^{\infty}\bigg[\sqrt[n]{n}-1\bigg]^n$"

Applying the root D'alembert's root test we get

$\displaystyle \lim_{n\to\infty}\bigg[\sqrt[n]{n}-1\bigg]^{\frac{n}{n}}=\lim_{n\to\infty}\sqrt[n]{n}-1=0$

$\displaystyle \therefore$ series is convergent? Is there soemthing wrong witht that?

Secondly

I dont know what to call this...is this actually a formal definition...I am about 99% sure its correct

Let $\displaystyle \sum_{n=0}^{\infty}a_n=\sum_{n=0}^{\infty}b_n\pm{c _n}$

If either $\displaystyle \sum_{n=0}^{\infty}b_n$ or $\displaystyle \sum_{n=0}^{\infty}c_n$ diverges we may say that $\displaystyle \sum_{n=0}^{\infty}a_n$ diverges

Is it sufficent to say that as a justification?