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Math Help - real roots of function

  1. #1
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    real roots of function

    How do I show that the function

    fn(x) = 1 + x + (x^2)/2! + ... + (x^n)/n!

    has no real roots when n is an even number,
    and has one real root when n is an odd number?
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  2. #2
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    Quote Originally Posted by weasley74 View Post
    How do I show that the function

    fn(x) = 1 + x + (x^2)/2! + ... + (x^n)/n!

    has no real roots when n is an even number,
    and has one real root when n is an odd number?
    Suppose that f_n(x) has the property that for any even n we have no real roots. We can easy show that f_n(x) has property that is has exactly one real root. Note that f'_n(x) = f_{n-1}(x) where n-1 is even, and so the derivative is non-vanishing for f_n(x), thus it is an increasing odd polynomial, thus, it has exactly one root.

    There is an elementary way to prove that f_n(x) > 0 for n even, which will complete what you want to show. Using this we can prove this. Let F(x) = \frac{x^n}{n!} then F(x)\geq 0 if n is even. This mean f_n(x) = F(x)+F'(x)+...+F^{(n)}(x) \geq 0.

    I will try to come up with an approximating solution to this problem also, since I like putting bounds on functions.
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  3. #3
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    Here is an approximation-type proof. First if x>0 then certainly f_n(x) > 0 and if x=0 then f_n(x)\not = 0. Thus, if is safe to assume that x<0. Let f(x) = e^x by Taylor's theorem it means for any x<0 we have e^x - f_n(x) = \frac{e^y}{(n+1)!} x^{n+1} where x<y<0 because f_n(x) is a Taylor n degree polynomial. Thus, f_n(x) = e^x - \frac{e^y}{(n+1)!}x^{n+1}. But e^y>0 \mbox{ and }x^{n+1} < 0 since  n is even and x<0. Thus, e^x - \frac{e^y}{(n+1)!}x^{n+1} > e^x > 0. This means, f_n(x) > 0 for all x\in \mathbb{R} and n even.
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  4. #4
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    Isn't therse an easier way to prove this, cause frankly I don't really get this.. maybe i'm just dumb..?
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  5. #5
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    Quote Originally Posted by weasley74 View Post
    Isn't therse an easier way to prove this, cause frankly I don't really get this.. maybe i'm just dumb..?
    Are you familar with Taylor's theorem because that is all you need to know.
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