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Math Help - Period

  1. #1
    Senior Member pankaj's Avatar
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    Period

    If \sum_{r=0}^{21}f(\frac{r}{11}+2x)=constant for all real x and f(x) is periodic ,then what could be period of f(x)
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by pankaj View Post
    If \sum_{r=0}^{21}f(\frac{r}{11}+2x)=constant for all real x and f(x) is periodic ,then what could be period of f(x)
    Hint you know that cos is periodic function right it is periodic with period 2\pi right

    see this

    h(x)=cosx = cos (x+2\pi) = cos(x+4\pi) = cos(x+6\pi ) = cos(x+8\pi)

    so h(x) periodic with period 2\pi

    ok cos(2x) periodic with period ?? \pi right

    cos(2x) = cos(2x+\pi) = cos (2x+2\pi ) = cos(2x+3\pi)= ...

    dose this make a sense .

    did you got it
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  3. #3
    Senior Member pankaj's Avatar
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    Taking example is allright but we need to generalize .

    If we replace x by x+\frac{1}{22} then see what happens.

    I am going to bed.It is midnight here.
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  4. #4
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by pankaj View Post
    If \sum_{r=0}^{21}f(\frac{r}{11}+2x)=constant for all real x and f(x) is periodic ,then what could be period of f(x)
    f(2x)+f(2x+\frac{1}{11}) + f(2x+\frac{2}{11})+f(2x+\frac{3}{11})+...f(2x+\fra  c{21}{11})=c ...(1)

    f(2x+\frac{1}{11})+f(2x+\frac{2}{11})+f(2x+\frac{3  }{11})+....f(2x+\frac{22}{11}) = c...(2)


    find the subtraction of (2)-(1)

    f(2x)-f(2x+2)=0

    f(2x)=f(2x+2)

    so f(2x) is peroid with period 2 and
    f(x) is periodic with period 4 right
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  5. #5
    Senior Member pankaj's Avatar
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    Quote Originally Posted by Amer View Post
    [tex]

    f(2x)-f(2x+2)=0

    f(2x)=f(2x+2)

    so f(2x) is peroid with period 2 and
    f(x) is periodic with period 4 right

    Actually if you replace x by \frac{x}{2} then we have
    f(x)=f(x+2)
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  6. #6
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by pankaj View Post
    Actually if you replace x by \frac{x}{2} then we have
    f(x)=f(x+2)


    is there a different between the periods of f(x) and f(2x)


    ok lats do you said replace x with x/2


    f(x)+f(x+\frac{1}{11})+f(x+\frac{2}{11})+f(x+\frac  {3}{11})+...+f(x+\frac{21}{11})=c

    sub instead of x x+1/11

    f(x+\frac{1}{11})+f(x+\frac{2}{11})+f(x+\frac{3}{1  1})+f(x+\frac{4}{11})...+f(x+\frac{22}{11})=c

    f(x)=f(x+2) ??

    ok sub instead of x 3x/2

    f(3x)+f(3x+\frac{1}{11})+f(3x+\frac{2}{11})+...+f(  3x+\frac{21}{11})=c

    instead of 3x 3x+1\11

    f(3x+\frac{1}{11})+f(3x+\frac{2}{11})+...f(3x+\fra  c{22}{11})=c

    f(3x)=f(3x+2) ??

    so the periodic dose not change for all values of x so we can say

    f(nx)=f(nx+2)..n\in R right
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  7. #7
    Senior Member pankaj's Avatar
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    there is a well known proven result that if period of f(x) is T,then period of f(ax+b) is given by \frac{T}{|a|} where a\neq 0
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