Suppose $$\sum_{n=1}^{\infty}a_{n} = L_{1}, \sum_{n=1}^{\infty}b_{n} = L_2 $$ Is $$\sum_{n=1}^{\infty}a_{n}b_{n} = L_{1}L_{2}$$ correct? If not, is there any counterexample?
2026-03-31 18:55:04.1774983304
Is there a counterexample such that $\sum_{n=1}^{\infty}a_nb_n\ne \left(\sum_{n=1}^{\infty}a_n\right)\left(\sum_{n=1}^{\infty}b_n\right)$?
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In general, $$a_1 b_1 + a_2 b_2 \ne (a_1+a_2)(b_1+b_2).$$