The number of times the function vanishes

[utkarshakash]

Homework Statement

The number of times the function $\sum_1^{2009} \dfrac{r}{x-r}$ vanishes is

Homework Equations

The Attempt at a Solution

Expanding the sum and writing first few terms of it

$\frac{1}{x-1} + \frac{2}{x-2} + \frac{3}{x-3} ...$

Now if I take the LCM I will get a polynomial of degree 2008 in the numerator and (x-1)(x-2)...(x-2009) in the denominator. f(x) vanishes when the num becomes 0. In other words I have to find roots of the num. But how can I be sure that all its 2008 roots will be real?

 

[Office_Shredder]

You might want to consider what happens when x is very close to an integer, and what that implies about the existence of roots.

 

[gopher_p]

The Intermediate Value Theorem could be useful.

 

[utkarshakash]

You might want to consider what happens when x is very close to an integer, and what that implies about the existence of roots.

When x is very close to an integer lying between 1 and 2009 denominator tends to zero and f(x) tends to infinity. How does this help me?

 

[Office_Shredder]

No, it doesn't always tend to infinity, sometimes it tends to something else... Think about the graph of 1/x

 

[brmath]

If x < 1 this sum consists entirely of negative terms, so will have not zeros in that range. Suppose n < x < n+1. Then all the terms up to n/(x-n) will be positive and all the rest will be negative. It seems that there may be an x where the positive and negative terms cancel.

If there is such an x, it must be the only one, because making it larger will make create more positive terms and fewer negative terms, and making it smaller vice versa. Can you formalize a way to say this?

Could it be that there is no such x? Why or why not?