Question

I have a complex sequence: \[\left\{ a_n \right\}_{n=1}^\infty,a_n \neq0\forall n\] and \[\lim_{n \rightarrow \infty}a_n=0\]. Now I need to show that \[\sum_{n=1}^{\infty}a_1a_2a_3...a_n \] is absolute convergent. I am currently just stuck, any suggestions?

Answer 1

Can I use the ratio test: \[q=\lim_{n \rightarrow \infty}\frac{ \left| a_{n+1} \right| }{ \left| a_n \right| }\] and check if q>1, or q<1? Or does this test not apply for complex sequences?

Answer 2

You can use ratio test.