[Not HW]
Let $\left(a_{n}\right)_{n=1}^{\infty}$ a sequence of real numbers such that, for all $ n\in\mathbb{N}$ , $0
Why the following statements don't imply the convergence of $\sum_{n=1}^{\infty}a_{n}$ :
$\lim_{n\to\infty}n^{n\cdot a_{n}}=1$
and the second one:
$\frac{a_{n+1}}{a_{n}}<1-\frac{1}{n+1}$
For the second one, since the ratio test calls for the $\lim_{n\to\infty}\frac{a_{n+1}}{a_{n}}$ to be strictly less than 1, it's obvious that the above statement does not imply convergence, but I didn't find a counterexample neither.
At the exam, I did mark statement 1 as a sufficient condition for convergence, but it's not true. So we need to find $a_{n}$ such that:
- $\lim_{n\to\infty}n^{n\cdot a_{n}}=1$ $\iff$ $\lim_{n\to\infty}n\cdot a_{n}\ln n=0$
- $0
- $\sum_{n=1}^{\infty}a_{n}$ diverges.
I thought of the following sequence:
$a_{n}=\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{16}\dots$
$\sum a_{n}$ clearly diverges but I'm not sure about $\lim_{n\to\infty}n^{n\cdot a_{n}}$ .
Thanks guys for your help.