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Let be [itex]\Sigma_{n=1}^{\infty} a_n[/itex] a series in ℝ .Suppose that [itex]\Sigma_{n=1}^{\infty} a_n[/itex] is absolutely convergent . Suppose that for each Q [itex]\in[/itex] N , [itex]\Sigma_{n=1}^{\infty} \frac{a_n}{Q^n}[/itex] is convergent and [itex]\Sigma_{n=1}^{\infty} \frac{a_n}{Q^n} = 0[/itex].Then [itex]a_n = 0[/itex] for all n [itex]\in[/itex] N.

My second question is : How Can I view direct proportionality rigorously without referring to the terms which are not precise like " Varying quantities " , " Variable " ? I tried doing this and I reached some ideas like making this variable as the value of a function. Is there a definitions all mathematicians work with different from that which says that y varies directly as x if there is a constant k , where y = kx ?, here we can't consider variable and constant as mathematical terms , because each element of set is a single element , but we use variable as a conventions only in writing as the value of a function for example in order to make writing easy , so Is there a definitions all mathematicians work with ?

Thanks .

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# Is that true ?

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