Integral to Infinity of Reciprocal of Power of x

Theorem
The improper integral


 * $\displaystyle \int_1^\infty \dfrac {\d t} {t^x}$

exists $x > 1$.

Proof
First let $x \ne 1$.

Then:

If $x > 1$, then $x - 1 > 0$.

Hence from Power of Reciprocal, $\dfrac 1 {P^{x - 1} } \to 0$ as $P \to +\infty$.

If $x < 1$, then $x - 1 < 0$.

Hence $P^{x - 1} \to 0$ as $P \to +\infty$.

Then from Reciprocal of Null Sequence it follows that $\dfrac 1 {P^{x - 1} } \to \infty$ as $P \to +\infty$.

Finally we have that from Integral of Reciprocal is Divergent:


 * $\displaystyle \lim_{P \mathop \to \infty} \int_1^P \dfrac {\d t} t \to \infty$ as $P \to \infty$.

All cases are covered, and the result follows.