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Is that true ?

  1. Jan 10, 2012 #1

    MIB

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    Is that true ?

    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 .
     
    Last edited: Jan 10, 2012
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  3. Jan 11, 2012 #2

    micromass

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    Hello MIB!!

    Do you know some things about power series and analytic functions?
     
  4. Jan 11, 2012 #3

    MIB

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    yes , In fact I am using this to prove the uniqueness of the representations of functions as a power series , I yes I know about power series and series of functions and sequences of functions ... etc
     
  5. Jan 11, 2012 #4

    micromass

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    Good. Do you know the theorem of "unique analytic continuation"?? That is: if X is a set with an accumulation point and if [itex]\sum{a_nx^n}=0[/itex] for all [itex]x\in X[/itex], then [itex]a_n=0[/itex]??

    Do you see what to take as X here??
     
  6. Jan 11, 2012 #5

    MIB

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    no , I don't know it , but if I knew , I would put X as the set of 1/Q , where Q is a natural number and the number 0 , and the accumulation point is 0
     
    Last edited: Jan 11, 2012
  7. Jan 11, 2012 #6

    MIB

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    ok must I begin know to prove the generalized theorem with this background in Mathematics ?
     
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