Limit Sum Riemann: Solve Homework Problem

[Molina]

Homework Statement

Calculate the limit with Riemann.

Homework Equations

$$\displaystyle\lim_{n \to{+}\infty}{\displaystyle\frac{pi}{4}\cdot{} \displaystyle\sum_{k=0}^n{tan^2(\displaystyle\frac{k\cdot{} pi}{4n})\cdot{}\displaystyle\frac{1}{n}}}$$

The Attempt at a Solution

I don't know how to start this problem...

Help me please, thank you.

 

[HallsofIvy]

"With Riemann" means that you treat this as a Riemann sum- which is used to define the definite integral.

If we divide the x-axis, from x= a to x= b, into n intervals then each interval has length [itex]\Delta x= (b- a)/n[itex]. And, if we take the value of the integrand, f, in each interval at the left end of that interval, we have f((b-a)k/n) as the height of the rectangle we are forming on that interval so its area is f((b-a)k/n)(b-a)/n and the whole arera is
[tex]\displaystyle (b- a)\sum_{k=0}^n f((b-a)k/n)\cdot\frac{1}{n}[tex]
The limit turns that into an integral.

So you need to indentify f(x), a, and b in this particular sum, and integrate.

 

[Molina]

Explain me in this case please, is the first time I have to do one problem like this. Thank you

 

[Gib Z]

Halls' already explained it, you just have to put in a bit of effort into trying to understand what he said. Some of his code didn't appear properly so I'll put it up again for you.

"With Riemann" means that you treat this as a Riemann sum- which is used to define the definite integral.

If we divide the x-axis, from $x= a$ to $x= b$, into $n$ intervals then each interval has length $\Delta x= (b-a)/n$. And, if we take the value of the integrand, $$f(x)$$, in each interval at the left end of that interval, we have $$f\left(\frac{(b-a)k}{n}\right)$$ as the height of the rectangle we are forming on that interval so its area is $$f(\frac{(b-a)k}{n})\left(\frac{b-a}{n}\right)$$ and the whole area is
$$\displaystyle (b- a)\sum_{k=0}^n f\left( \frac{(b-a)k}{n}\right)\cdot\frac{1}{n}$$
If we take the limit as the number of rectangles $n \to \infty$ this becomes $$\int^b_a f(x) dx$$.
So you need to indentify f(x), a, and b in this particular sum, and integrate.

So carry you this idea for an easy (and common) case, $$\int^1_0 f(x) dx$$. You want to find an expression for Riemann sums for that integral. If you do the same idea as what Halls described already, you should get $$\int^1_0 f(x) dx = \lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n}\right)$$.

 

[Molina]

I'm sorry but I don't understand, how i know a and b? I'm sorry I'm lost.

I have the idea.. b is $$pi / 4$$ and $$a = 0$$ ?

And f(x) = $$tan^2(k*pi/4n)$$ ? ?