1. ## convergences

-working in real numbers, true or false? proof or counterexample:
(a) If $\displaystyle a_n$ is decreasing sequence of positive numbers, and $\displaystyle na_n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$, then $\displaystyle \sum a_n$ converges.

(b)If $\displaystyle \sum a_n$ converges, then $\displaystyle \sum \frac{a_n}{\sqrt{n}}$converges.

(c)If $\displaystyle \sum a_n$ converges, then $\displaystyle \sum \frac{|a_n|}{n}$ converges.

(d)If $\displaystyle \sum a_n$ converges, then $\displaystyle \sum \frac{|a_n|}{n^{3/2}}$converges.

2. Originally Posted by DontKnoMaff
working in real numbers, true or false? proof or counterexample:
(a) If $\displaystyle a_n$ is decreasing sequence of positive numbers, and $\displaystyle na_n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$, then $\displaystyle \sum a_n$ converges.
Consider:

$\displaystyle a_n=\frac{1}{n\ln(n)}$

$\displaystyle \displaystyle \lim_{n\to \infty} na_n=\frac{1}{\ln(n)}=0$

but: $\displaystyle \sum_{n=2}^{\infty}a_n$ diverges (you might have an interesting time proving this )

CB

3. could compare it to $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$.

4. Originally Posted by Time
could compare it to $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$.
NO since $\displaystyle \frac{1}{n\ln(n)}<\frac{1}{n}$ unless you have something mor sophisticated in mind.

CB