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Math Help - Proffs - various integration proffs

  1. #1
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    Proffs - various integration proffs

    Problem 1 :

    Let f:[1,∞[ R be a positive continuous decreasing function. Show if m,n N and m<n that the integral

    ∫ f(x) dk from (m+1) to (n+1) ≤∑ f(k) (sum from k=1 to n) -∑ (sum from k=1 to m) ≤f(m)+∫f(x)dk (with integral over the interval from m to n)


    Problem 2 :

    From problem 1 prove if ∫ f(x) dk on the interval from 1 to infinity is convergent that the integral ∫ f(x) dk from (m+1) to (infinity) ≤∑ f(k) (sum from k=1 to infinity) -∑ (sum from k=1 to m) ≤f(m)+∫f(x)dk (with integral over the interval from m to infinity)


    Problem 3 :
    Let snm = =∑ 1/k (sum from k=1 to nm) - (sum from k=1 to n times m)
    Show from problem 1 if sn=∑ 1/k (sum from k=1 to n) for all n N that the sum difference limit : lim (snm - sm) = ln(n)


    Problem 4 :

    Proff that the functional serie fn given by fn : R R for n=1,2,3 ..... given by

    fn(x) = ____1________
    1+ exp((n(2-x))

    is pointwise convergent and find the limiting function.

    Is the functional serie also uniform convergent on R ? Argue for your result.


    Help proving these four problems is urgently needed.

    Thanks
    Pragma
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  2. #2
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    Quote Originally Posted by Pragma View Post
    Problem 1 :

    Let f:[1,∞[ R be a positive continuous decreasing function. Show if m,n N and m<n that the integral

    ∫ f(x) dk from (m+1) to (n+1) ≤∑ f(k) (sum from k=1 to n) -∑ (sum from k=1 to m) ≤f(m)+∫f(x)dk (with integral over the interval from m to n)


    Problem 2 :

    From problem 1 prove if ∫ f(x) dk on the interval from 1 to infinity is convergent that the integral ∫ f(x) dk from (m+1) to (infinity) ≤∑ f(k) (sum from k=1 to infinity) -∑ (sum from k=1 to m) ≤f(m)+∫f(x)dk (with integral over the interval from m to infinity)


    Problem 3 :
    Let snm = =∑ 1/k (sum from k=1 to nm) - (sum from k=1 to n times m)
    Show from problem 1 if sn=∑ 1/k (sum from k=1 to n) for all n N that the sum difference limit : lim (snm - sm) = ln(n)


    Problem 4 :

    Proff that the functional serie fn given by fn : R R for n=1,2,3 ..... given by

    fn(x) = ____1________
    1+ exp((n(2-x))

    is pointwise convergent and find the limiting function.

    Is the functional serie also uniform convergent on R ? Argue for your result.


    Help proving these four problems is urgently needed.

    Thanks
    Pragma

    Too many questions, too little self work shown: what have you done so far? Where are you stuck? These are questions at least from advanced calculus I or calculus II, so you must know something before attacking this stuff: show us!

    Tonio
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