hi
Test the series for convergence:
$\displaystyle \sum_{n=1}^{\infty} \left( 1 - \sqrt[n]{n} \right)$
am sure, it will be tested by using one of the comparison tests
but i can't handle it
I think I have an answer that works, but I'm not 100% sure.
We know that $\displaystyle \sum_{n=1}^{\infty} \left( \frac{1}{n} \right)$ diverges.
I believe that, for n > 2, it can be shown that $\displaystyle \sqrt[n]{n} - 1 > \frac{1}{n}$. Thus $\displaystyle \sum_{n=1}^{\infty} \left( 1 - \sqrt[n]{n} \right)$ diverges.
I tried it as a guess, and so I worked backwards from the desired result:
$\displaystyle \sqrt[n]{n} - 1 > \frac{1}{n}$
$\displaystyle \sqrt[n]{n} > 1 + \frac{1}{n}$
$\displaystyle \sqrt[n]{n} > \frac{n+1}{n}$
$\displaystyle n > \left( \frac{n+1}{n} \right)^n$
$\displaystyle n > \frac{(n+1)^n}{n^n}$
$\displaystyle n^{n+1} > (n+1)^n$
It seems like the last statement holds true for all n > 2, and the steps could simply be written in reverse order to prove what we wanted to prove.
To prove this last inequality, you can say that the right hand side $\displaystyle \left(1+\frac 1n\right)^n$ converges toward $\displaystyle e$. In particular, it is bounded, and hence less than $\displaystyle n$ for large $\displaystyle n$.
Alternatively, from the beginning, one can write $\displaystyle \sqrt[n]{n}-1=e^{\frac{\log n}{n}}-1\sim\frac{\log n}{n}$ so if you know the comparison test using asymptotic equivalence, you can conclude using $\displaystyle \frac{\log n}{n}\geq \frac{1}{n}$ and the divergence of $\displaystyle \sum_n \frac{1}{n}$.
Thanks for your explanation. I should have noticed the connection with $\displaystyle e$!
Regarding the asymptotic solution, would you mind explaining how you got $\displaystyle e^{\frac{\log n}{n}}-1\sim\frac{\log n}{n}$ ? I can verify that the LHS divided by the RHS tends to 1 as n tends to infinity, but I don't know why.
You agree that $\displaystyle e^u-1\sim u$ as $\displaystyle u\to0$ (just another way of writing $\displaystyle \frac{e^u-1}{u}\to 1$, which is the derivative $\displaystyle \exp'(0)=1$). Furthermore, $\displaystyle \frac{\log n}{n}\to 0$ as $\displaystyle n\to\infty$. We may therefore compose limits and deduce $\displaystyle e^{\frac{\log n}{n}}-1\sim\frac{\log n}{n}$ as $\displaystyle n\to\infty$. Here it is!