convergences

• Oct 20th 2010, 10:44 PM
DontKnoMaff
convergences
-working in real numbers, true or false? proof or counterexample:
(a) If $a_n$ is decreasing sequence of positive numbers, and $na_n \rightarrow 0$ as $n \rightarrow \infty$, then $\sum a_n$ converges.

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

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

(d)If $\sum a_n$ converges, then $\sum \frac{|a_n|}{n^{3/2}}$converges.
• Oct 21st 2010, 12:19 AM
CaptainBlack
Quote:

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

Consider:

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

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

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

CB
• Oct 21st 2010, 10:47 AM
Time
could compare it to $\sum_{n=1}^{\infty} \frac{1}{n}$.
• Oct 21st 2010, 02:11 PM
CaptainBlack
Quote:

Originally Posted by Time
could compare it to $\sum_{n=1}^{\infty} \frac{1}{n}$.

NO since $\frac{1}{n\ln(n)}<\frac{1}{n}$ unless you have something mor sophisticated in mind.

CB