convergences

**DontKnoMaff** · October 20th 2010, 10:44 PM

-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.

**CaptainBlack** · October 21st 2010, 12:19 AM

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

)

CB

**Time** · October 21st 2010, 10:47 AM

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

**CaptainBlack** · October 21st 2010, 02:11 PM

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

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