Evaluate lim(n→∞)(1+2+3+...+n)/n^2

Question

Evaluate: 

\(\lim\limits_{n \to ∞}\frac{1+2+3+.....+n}{n^2}\)

Answer 1

\(\lim\limits_{n \to ∞}\frac{1+2+3+.....+n}{n^2}\) 

\(=\lim\limits_{n \to ∞}\frac{\frac{n(n+1)}{2}}{n^2}\) 

\(=\frac{1}{2}\lim\limits_{n \to ∞}\frac{n^2+n}{n^2}\) 

\(=\frac{1}{2}\lim\limits_{n \to ∞}(1+\frac{1}{n})\) 

\(=\frac{1}{2}(1+0)\) 

\(=\frac{1}{2}\times1\) 

\(=\frac{1}{2}\)