# The number of times the function vanishes

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## Homework Statement

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

## 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?

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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.

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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?

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