\(\frac{k^{r}}{\left ( 1+p \right )^{k}}\to 0\).

Let \(\displaystyle \lim_{k \to \infty }\frac{k^{a}}{b^{k}}\to 0\)

And \(a> 0, b> 1 \) ......(a)

Now, take the limit \(\frac{k^{r}}{\left ( 1+p \right )^{k}}\) as \(k\to \infty\)

And \(\left ( r,p \right )> 0\epsilon \mathbb{R}\).

By this, we get three cases:

Case 1: \(r=0\) then \(k^{0}=1\)

So, when \(k\to \infty , \left ( 1+p \right )^{k}\to \infty\)

So, \(\frac{k^{r}}{\left ( 1+p \right )^{k}} \to 0\) when \(k\to \infty\)

Now, Case 2: When \(r<0\) then \(k^{r}\to 0\)

So, when \(k\to \infty , \left ( 1+p \right )^{k}\to \infty\)

So, \(\frac{k^{r}}{\left ( 1+p \right )^{k}} \to 0\) when \(k\to \infty\)

Case 3: When \(r>0\)

We will use \(a=r\), \(b=1+p\) in equation (a),

\(\frac{k^{r}}{\left ( 1+p \right )^{k}} \to 0\) when \(k\to \infty\)