Integral of 1/ln(x). Convergence test

In summary, Some functions, such as 1/x, have straightforward integrals, but taking the inverse of them can make them more complicated. The integral of ln(x) diverges from 1 to infinity, regardless of the value of n. For values of n between 0 and 1, the integral of ln(x) still diverges, but at a slower rate. It is possible to show that ##\ln^n(x)<x## asymptotically, but further exploration is needed.
  • #1
0kelvin
50
5
Homework Statement
Integral of ln(x) from 1 to infinite diverges. But how do I know if the 1/ln(x) will diverge too?
Relevant Equations
1/ln(x)
Some functions have straight foward integrals, but they get complicated if you take the inverse of it. 1/f(x) for instance.

The primitive of 1/x is ln(x). In this case it's easy to check that the integral of 1/x or ln(x) from 1 to infinite diverges.

##\int_1^\infty (\ln(x))^n dx##

If n = 0, I have f(x) = 1. This cannot converge.

If n = 1, I have that the integral diverges.

If n < 0, then I have no idea except to let wolfram tell me.

If 1 < n < 0, the integral of ln(x) already diverges, taking the root of it just slows down a bit but still diverges.
 
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  • #2
For 1 < x you have ln(x) < x ##\Rightarrow## 1/ln(x) > 1/x.
If the integral of 1/x diverges, the integral of 1/ln(x) certainly diverges ...
 
  • #3
BvU said:
For 1 < x you have ln(x) < x ##\Rightarrow## 1/ln(x) > 1/x.
If the integral of 1/x diverges, the integral of 1/ln(x) certainly diverges ...
... and the next challenge is to show that asymptotically ##\ln^n(x)<x##.
 
  • #4
haruspex said:
... and the next challenge is to show that asymptotically ##\ln^n(x)<x##.
If I guessed correctly, would this be by finding this : ##\frac{n!}{x^n}=0## at ##\infty##?
 
Last edited:

FAQ: Integral of 1/ln(x). Convergence test

1. What is the integral of 1/ln(x)?

The integral of 1/ln(x) is a logarithmic integral function, often denoted as Li(x). It is a special function that cannot be expressed in terms of elementary functions, but can be approximated through numerical methods.

2. How is the integral of 1/ln(x) calculated?

The integral of 1/ln(x) can be calculated using various methods such as substitution, integration by parts, or using complex analysis techniques. However, there is no closed-form solution for the integral, so it is often approximated through numerical methods.

3. Does the integral of 1/ln(x) converge or diverge?

The integral of 1/ln(x) is known to converge for values greater than 1, but diverges for values less than or equal to 1. This can be mathematically proven using the convergence test for improper integrals.

4. Can the convergence of the integral of 1/ln(x) be tested?

Yes, the convergence of the integral of 1/ln(x) can be tested using the convergence test for improper integrals. This involves determining the behavior of the function as the upper limit of integration approaches a certain value, typically infinity.

5. What is the significance of the integral of 1/ln(x) in mathematics?

The integral of 1/ln(x) is a special function that arises in various mathematical and scientific contexts. It has applications in number theory, probability, and physics. It also has connections to the Riemann zeta function and the prime number theorem.

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