# Evaluate the limit of a series with an integral

1. Aug 9, 2013

### e^(i Pi)+1=0

1. The problem statement, all variables and given/known data

$\lim_{n \to \infty} \sum_{i=1}^{n} \frac{4}{n}\sqrt{\frac{4i}{n}}$

3. The attempt at a solution

This seems to give the right answer, 16/3, but I can't figure out why:

$\lim_{n \to \infty}\int_{1}^{n}\frac{4}{n}\sqrt{\frac{4x}{n}}dx$

2. Aug 9, 2013

### vanhees71

I'd rather think that you have to interpret the sum as an approximation to the (Riemann) integral
$$I=\int_0^{4} \mathrm{d} x \sqrt{x},$$
where the interval is divided in $n$ subintervals of equal size.

3. Aug 9, 2013

### HallsofIvy

This is more than just approximating. If you have the function $f(x)= \sqrt{x}$, on the interval [0, 4], the Riemann sum, dividing [0, 4] into n equal subintervals, so that each subinterval has length 4/n and x= 4i/n, gives $$\sum{i= 1}^n \frac{4}{n}\sqrt{\frac{4i}{n}}$$. As n goes to infinity, we are dividing the interval into more and more smaller and smaller intervals and the limit is the Riemann integral.

4. Aug 9, 2013

### haruspex

vanhees71 said the sum approximates the Riemann integral; you're saying the limit of the sum is the Riemann integral. No contradiction there.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted