The number of times the function vanishes

• utkarshakash
In summary, the function \sum_1^{2009} \dfrac{r}{x-r} has 2008 roots that are real and distinct. The existence of these roots can be proven using the Intermediate Value Theorem and considering the behavior of the function when x is very close to an integer. It is possible that there is only one root, as making x larger or smaller will result in either more positive terms or more negative terms, respectively. It is also possible that there are no roots, but this would require further investigation.

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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?

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

The Intermediate Value Theorem could be useful.

Office_Shredder said:
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?

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

1 person
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?

What is the meaning of "the number of times the function vanishes"?

The number of times the function vanishes refers to the number of points on a graph where the function's output, or y-value, is equal to zero.

Why is it important to know the number of times a function vanishes?

Knowing the number of times a function vanishes can provide important information about the behavior of the function, such as the number of solutions to an equation or the number of roots in a polynomial.

How can I determine the number of times a function vanishes?

The number of times a function vanishes can be determined by graphing the function and counting the number of points where the graph crosses the x-axis, or by solving the equation f(x) = 0.

What is the relationship between the degree of a polynomial function and the number of times it vanishes?

The degree of a polynomial function is equal to the highest power of x in the function. The number of times the function vanishes is equal to the degree of the polynomial function. For example, a polynomial function with a degree of 3 can vanish up to 3 times.

Can a function vanish an infinite number of times?

Yes, a function can vanish an infinite number of times. This is known as a function with infinitely many roots, or a function that has no real solutions. An example of this is the function f(x) = sin(x), which vanishes an infinite number of times as it crosses the x-axis an infinite number of times.