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Question Number 28977 by abdo imad last updated on 02/Feb/18
$${let}\:{give}\:{u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{a}_{{k}} ^{\mathrm{2}} \:\:\:\:\:{with}\:\left({a}_{{k}} \right)\:{sequence}\:{of}\:{reals}/{a}_{{k}>\mathrm{0}} \\ $$ $${and}\:{v}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{a}_{{k}} }{{k}}\:.\:{prove}\:{that}\:{u}_{{n}} {converges}\Rightarrow\left({v}_{{n}} \right){converges} \\ $$