Hi i have been trying to prove this summation from i=1 to n is less than $\frac{1}{(1-r)^3}$. The only method i could think of is by solving with a geometric series.But i am unsure of the proofing . Please advice ..

$$\sum\_{i=1}^n (1+2+3+...+i)r^i < \frac{1}{(1-r)^3}$$
for all $n\ge1$ and $0\lt r\lt1$

using the fact that $$\sum\_{i=1}^n ir^i < \frac{r}{(1-r)^2}$$
for all $n\ge1$ and $0\lt r\lt1$