# Proving the divergent integral of 1/f(x) as x-> infinity

• 000
In summary, the conversation discusses the existence of a function f(x) such that the indefinite integral of 1/f(x) as x approaches infinity diverges, and f(x) is always greater than or equal to x. It is then questioned whether this function must be a linear polynomial. While there are many non-polynomial functions that satisfy these conditions, it is true that if f(x) is required to be a polynomial, it must be linear. The possibility of an asymptotic function that is always larger than x is also considered, with an example of f(x)=x+1/x being suggested.
000

## Homework Statement

There exists a function f(x) such that the indefinite integral of 1/f(x) as x-> infinity diverges, and f(x) >= x for all values of x. Prove this function must be a linear polynomial.

## Homework Equations

None that I know of.

## The Attempt at a Solution

No idea where to start.

Last edited:
You can't, as stated, it is not true. There exist many non-polynomial functions that are "asymptotic" to x such that the integral of 1/f(x) diverges. However, if you require that f(x) be a polynomial, then it is true that f must be linear.

HallsofIvy said:
You can't, as stated, it is not true. There exist many non-polynomial functions that are "asymptotic" to x such that the integral of 1/f(x) diverges. However, if you require that f(x) be a polynomial, then it is true that f must be linear.

f(x) must always be larger than x, are there any asymptotic functions that fit that criteria?

000 said:
f(x) must always be larger than x, are there any asymptotic functions that fit that criteria?

## 1. What does it mean for an integral to be divergent?

An integral is said to be divergent if it does not have a finite value. In other words, the integral does not converge to a specific number and instead either increases or decreases without bound.

## 2. Why is it important to prove the divergence of an integral?

Proving the divergence of an integral is important because it helps to understand the behavior of a function as the independent variable approaches infinity. This information is crucial in many areas of mathematics and science, such as in calculating limits and determining the convergence of series.

## 3. What is the general approach to proving the divergence of an integral?

The general approach to proving the divergence of an integral is to show that the integral either increases or decreases without bound as the independent variable approaches infinity. This can be done by using various techniques, such as comparison tests, limit comparison tests, or the integral test.

## 4. Is there a specific method for proving the divergence of 1/f(x) as x-> infinity?

Yes, there is a specific method for proving the divergence of 1/f(x) as x-> infinity, known as the limit comparison test. This method involves comparing the given integral to a known divergent integral and using the limit of the ratio between the two to determine divergence.

## 5. Can the divergence of 1/f(x) as x-> infinity be proven using other tests?

Yes, the divergence of 1/f(x) as x-> infinity can also be proven using the integral test or the comparison test. However, the limit comparison test is often the most efficient and straightforward method for proving the divergence of this type of integral.

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