Calculate limit of a series

Question

Calculate
$$ \lim_{n \to \infty} \sum \limits_{k=1}^n \frac{n}{k^2 - 4n^2} $$

or prove it doesn't exist.

Answer 1

This is a Riemann sum:

$$\sum_{k=1}^n \frac{n}{k^2-4 n^2} = \frac{1}{n} \sum_{k=1}^n \frac{1}{(k/n)^2-4}$$

Thus the limit as $n \to \infty$ is

$$\int_0^1 \frac{dx}{x^2-4} = -\frac12 \int_0^{\pi/6} d\theta \, \sec{\theta} = -\frac14 \log{3}$$

Answer 2

Hint: You can use Riemann Sums. See here.