If we know that $\sum a_n$ converges, could we derive that $\sum \dfrac{a_n}{n}$ also converges? (Suppose we don't have $a_n \ge 0$) If not, please give an example~
Thanks a lot.
A little problem about convergence of series
0
$\begingroup$
sequences-and-series
convergence
2 Answers
1
The answer is yes. Use Dirichlet's Test. Note that if $\sum a_n$ converges then its partial sums are bounded and moroever the sequence $1/n$ is positive and decreasing.
1
This follows from Dirichlet's test (in the wikipedia article take $b_n=a_n$ and $a_n=1/n$).