Answer
Converges to $\dfrac{1}{p-1}$ when $p \gt 1$
Work Step by Step
Since, we have $\int_1^n \dfrac{1}{x^p} dx=[\dfrac{1}{-p+1}(x^{-p+1}]_1^n=\dfrac{1}{1-p}(\dfrac{1}{n^{p-1}-1})$
Consider $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{1}{1-p}(\dfrac{1}{n^{p-1}-1})$
Now, if $p \lt 1$ then $\dfrac{1}{n^{p-1}-1}$ approaches to $\infty$
if $p=1$ then we have $\dfrac{1}{1-p}(\dfrac{1}{n^{p-1}-1}) \to \dfrac{1}{0}$
if $p \gt 1$ then we have $\dfrac{1}{1-p}(\dfrac{1}{n^{p-1}-1}) \to \dfrac{1}{p-1}$
Hence, $\lim\limits_{n \to \infty} a_n=\dfrac{1}{p-1}$ and {$a_n$} is Convergent and converges to $\dfrac{1}{p-1}$ when $p \gt 1$
