- #1

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## Homework Statement

Write the integral that would define the mgf of the pdf,

[itex]f(x) = \frac 1{\pi} \frac 1{1+x^2} dx [/itex]

## Homework Equations

The moment generating function (mgf) is [itex] E e^{tX}[\itex].

## The Attempt at a Solution

My question really has to do with improper integrals. I must show the improper integral diverges:

[itex]\int_0^{\infty} e^{tx} \frac 1{\pi} \frac 1{1+x^2} dx [/itex].

Now if do integration by parts and let [itex]u=e^{tx}[/itex] and [itex] dV = \frac 1{1+x^2}dx [/itex], then I have:

[itex] \frac 1{\pi} e^{tx} arctan(x) |_0^{\infty} - \int_0^{\infty} \frac 1{\pi} t arctan(x) e^{tx} dx [/itex].

However I can see that [itex] arctan(x) e^{tx} \frac 1{\pi} |_0^{\infty} [/itex], will be ∞. So is this enough to show that the improper integral diverges?