Determine the convergence or divergence of f(n) = 100,000n / (1 + n²)

by RoRi

February 27, 2016

Consider the function $f(n)$ defined by

$f(n) = \frac{100000n}{1+n^2}.$

Determine whether the sequence $\{ f(n) \}$ converges or diverges, and if it converges find the limit.

We have

$f(n) = \frac{100000n}{1+n^2} = 100000 \left( \frac{n}{1+n^2} \right).$

Then,

$\lim_{n \to \infty} f(n) = 100000 \cdot \lim_{n \to \infty} \frac{\frac{1}{n}}{\frac{1}{n^2} + 1} = 0.$

Hence $\{ f(n) \}$ converges to 0.

Stumbling Robot