Prove that an absolutely convergent series of complex numbers converges. This means to prove that $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}\mathbf{}$converges ($\mathbf{}{\mathbf{a}}_{\mathbf{n}}$and ${\mathbf{b}}_{\mathbf{n}}$real) if $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}\mathbf{}$converges. Hint: Convergence of $\mathbf{\sum }\mathbf{(}{\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{)}$means that localid="1658823032447" $\mathbf{\sum }{\mathbf{a}}_{\mathbf{n}}\mathbf{}$and $\mathbf{\sum }{\mathbf{b}}_{\mathbf{n}}$ both converge. Compare $\mathbf{\sum }\mathbf{\left|}{\mathbf{a}}_{\mathbf{n}}\mathbf{\right|}$ and$\mathbf{}\mathbf{\sum }\mathbf{\left|}{\mathbf{b}}_{\mathbf{n}}\mathbf{\right|}$with $\mathbf{\sum }\sqrt{{{\mathbf{a}}_{\mathbf{n}}}^{\mathbf{2}}\mathbf{+}{{\mathbf{b}}_{\mathbf{n}}}^{\mathbf{2}}}$ , and use Problem 7.9of Chapter 1.

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