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Math Help - how does this series behave..

  1. #1
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    how does this series behave..

    i was tought this trick to make series converge to 0
    for example
    f_n (x) =1 ,when x belongs to [n,(n+1)]
    f_n (x) =0 when it doesnt
    so no matter what x value we have
    when n goes to infinity the value 1 section will "run away
    and it always converge to 0

    now i cant undestand if the following function is a such a function.
    f_n(x)=(n)^0.5 ,when x belongs to [0,1/n]
    f_n(x)=0 ,when it doesnt

    herewhen n goes to infinty
    i dont have a moving section like before

    here the section shrinks
    and static am i correct?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by transgalactic View Post
    i was tought this trick to make series converge to 0
    for example
    f_n (x) =1 ,when x belongs to [n,(n+1)]
    f_n (x) =0 when it doesnt
    so no matter what x value we have
    when n goes to infinity the value 1 section will "run away
    and it always converge to 0

    now i cant undestand if the following function is a such a function.
    f_n(x)=(n)^0.5 ,when x belongs to [0,1/n]
    f_n(x)=0 ,when it doesnt

    herewhen n goes to infinty
    i dont have a moving section like before

    here the section shrinks
    and static am i correct?
    But you do have a moving section. Given any positive number x the Archimedean principle furnishes us with a N\in \mathbb{N} such that \frac{1}{N}<x thus for N \le n\implies f_n(x)=0. Thus for any positive x we can see that \lim_{n\to\infty}f_n(x)=0.
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  3. #3
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    i cant understand how its moving as n goes to infinty

    what you said is just that it has a value of 0
    but i think that its because the section moving
    but because the section is static but shrinks till zero

    for n=2
    0<x<1/2


    for n=100
    0<x<1/100
    am i correct?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by transgalactic View Post
    i cant understand how its moving as n goes to infinty

    what you said is just that it has a value of 0
    but i think that its because the section moving
    but because the section is static but shrinks till zero

    for n=2
    0<x<1/2


    for n=100
    0<x<1/100
    am i correct?
    I haven't the slightest idea of what you mean. Given any positive x by the Archimedean principle I know that eventually \frac{1}{n}<x\implies x\notin \left[0,\frac{1}{n}\right]\implies f_n(x)=0.
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  5. #5
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    imagy a graph of that function
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Perhaps the validation of another member will convince you. Maybe I am, in fact, wrong. In which case another member will chide me and we can move on.
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