# Math Help - Limit

1. ## Limit

Hi, can someone evaluate this limit? I need it for a mathematical model, so, technically, I'm allowed to use technology (Mathematica gives the result as approximately 23.7), but if someone knows an easy way to solve this, it would be better.

$\lim_{n \to \infty} \frac{90n}{\displaystyle\sum_{i=0}^{4n} 0.975^ \frac{i}{n}}=?$

Thanks.

2. I get the limit to be:

$-90 \frac{\ln 0.975}{1-0.975^4} = 23.6585 \dots$

The way I did this was to notice that the denominator is a geometric progression where $r = 0.975^{1/n}$ and so use the formula for the sum of a geometric progression to re-express the denominator.

Then, approaching the limit, the behaviour of the denominator is easy and the numerators' can be found by expanding in terms of a power series.

It's as easy as that!