TITLE: limits and summation
QUESTION [0 upvotes]: Possible Duplicate:
Riemann’s Integrals Question 

I have the following question,
$$ \lim_{n\to\infty} \sum_{i=1}^n \tan((\frac \pi {3n})i) \times \frac \pi {3n} $$  
and was wondering which limit laws I could use to work out the answer? this question 
is derived from a riemann integral question.
I know I can take out the $\frac \pi {3n} $   outside the summation so I'm left with 
$$ \lim_{n\to\infty} \frac \pi {3n} \sum_{i=1}^n \tan((\frac \pi {3n})i) $$  
but I'm not sure what to do next?

REPLY [0 votes]: As you have asked this question you already know the answer. Here is how you come up with $f(x)=\tan x$
Let \begin{equation}\mathcal{P}=\left\{ 0=x_0<x_1<...<\frac{i}{n}<...<x_n=1 \right\}\end{equation}
partition $[0,1]$
We want $$U_{f,\mathcal{P}}= \sum_{i=1}^n \tan((\frac \pi {3n})i)\frac \pi {3n} \iff \sum\limits_{i=1}^{n}\frac{\sup_{x\in [x_{{i-1}},x_i]}f(x)}n= \sum_{i=1}^n \tan((\frac \pi {3n})i)\frac \pi {3n} $$
If we choose an increasing function this simplifies to
$$\sum_{i=1}^n \tan((\frac \pi {3n})i)\frac \pi {3n} =\sum_{i=1}^{n}\frac{f(x_i)}n$$
Matching the terms gives
$$f(x_i)= \tan((\frac \pi {3n})i)\frac \pi {3}\Rightarrow f(x)= \tan((\frac \pi {3n})nx)\frac \pi {3}=\frac{\pi}3\tan(\frac{\pi x}3)$$
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