# If $\sum a_n b_n$ converges, does $\sum \overline{a_n} b_n$ converge too?

Let $$(a_n)$$ and $$(b_n)$$ be complex sequences. I don't think the following is true in general:

If $$\sum a_n b_n$$ converges, then $$\sum \overline{a_n} b_n$$ converges too.

Can someone provide a simple example to show that above statement is obviously false?

• can you think of two sequences where each term has the same modulus, but one converges and the other doesn't? Commented May 5, 2023 at 1:17
• @ZoeAllen Thank you, I can see clearly now. Commented May 5, 2023 at 5:42

Let $$a_n = b_n = \frac{i^n}{\sqrt n}$$, $$a_nb_n = \frac{(-1)^n}{n}$$ and $$\overline{a_n} b_n = \frac{1}{n}$$
Let $$a_{2n} = b_{2n} = \frac{1+i}{\sqrt{4n}}$$, $$a_{2n+1}=b_{2n+1} = \frac{1-i}{\sqrt{4n+2}}$$. Then $$a_nb_n = a_n^2 = (-1)^n\frac{i}{n}$$, so the series $$\sum a_nb_n = i\sum(-1)^n/n$$ converges by alternating series test, and $$\sum\overline{a_n}b_n = \sum|a_n|^2 = \sum 1/n$$ diverges.
• You need to add a $\sqrt{}$. Commented May 5, 2023 at 1:39