I need to show that the function $$f(x,y)=\frac{2}{n(n+1)},\,\, 1\leq y\leq x\leq n; 0 \text{ for other values of x and y}$$ where $n$ is a positive integer, is a density function for discrete random variable $(X,Y).$
To show $\sum_{x,y} f(x,y)=1,$ I tried to find $\sum_{x=1}^n\sum_{y=1}^x \frac{2}{n(n+1)}$ which does not turn out to be $1.$ Am I making some mistake in wrting the sum?
\begin{align} \sum_{x=1}^n \sum_{y=1}^x \frac{2}{n(n+1)} &= \frac{2}{n(n+1)} \sum_{x=1}^n \sum_{y=1}^x 1 \\&=\frac{2}{n(n+1)} \sum_{x=1}^n x\end{align}
Can you show that it is equal to $1$?