Question: Calculate  

$$ \lim\_{n \to \infty} \sum \limits\_{k=1}^n \frac{n}{k^2 - 4n^2} $$ 

or prove it doesn't exist.

Answer: 
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
> 
> 
> $$\int\_0^1 \frac{dx}{x^2-4} = -\frac12 \int\_0^{\pi/6} d\theta \, \sec{\theta} = -\frac14 \log{3}$$
> 
> 
>
**Hint:** You can use Riemann Sums. See [here](https://math.stackexchange.com/questions/472579/evaluating-sum-of-limits).