Taking an integral of a f(x) between 1 and infinity

[jared944]

Hi all!

I am integrating the function of 1/x between the values of x=1 and x=infinity. to do this, I am approximating the area of this function from 1 to 10, 1 to 100, 1 to 1000 (etc...). As I progress higher and higher, I noticed that the area stays fairly consistent after 1x10^17 (an approximate value of 33.0662578854).

So, I figure that the area of the function 1/x evaluated from one to infinity is not in fact infinity, but some finite number (around 33.066).

How can I truly express this? At first thought, I think I can integrate 1/x between 1 and the limit of N, as N approaches infinity. However, I would have no idea how to solve such a convoluted problem.

Any advice is appreciated! Thanks!

 

[galactus]

\(\displaystyle \L\\\int_{1}^{\infty}\frac{1}{x}dx\)

\(\displaystyle \L\\\lim_{n\to\infty}\int_{1}^{n}\frac{1}{x}dx\)

=\(\displaystyle \L\\\lim_{n\to\infty}\left[ln|x|\right]_{1}^{n}\)

=\(\displaystyle \L\\\lim_{n\to\infty}{ln|n|}={+\infty}\)

Thus, the integral diverges.

You can see it diverges.

For instance, if you use an upper limit of 1000. The result is 3ln(10).

If you use an upper limit of \(\displaystyle 1\times{10^{25}}\), the result is 25ln(10).

About 57.56.

The number of 0's equals the number in front of ln(10). It gets larger and larger. This is a rather hackneyed way to look at it, but you can see it isn't finite.

 

[tkhunny]

is not in fact infinity, but some finite number

"Infinity" is NOT a number.

\(\displaystyle \L\;\int_{1}^{10^{n}}{\frac{1}{x}}\;dx\;=\;n*ln(10)\)

Does that ending expression have a limit as n increases without bound?

You may be experiencing precision problems. Your arithmetic may be okay, but your practical implementation may be failing you.