Small angle forula problem

1. Mar 2, 2009

ghostyc

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

$$S_N(x)= \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\sin ((2 n-1)x)}{2 n-1}$$

By considering a suitable small angle formula show that the value of the sum at this point is

$$S_N \Big( \frac{\pi}{2 N} \Big)=\frac{2}{\pi} \int_0^{\pi} \frac{\sin (\mu)}{\mu} \; d{\mu}$$

2. Relevant equations

i have no idea how to get the suitable small angle formula working with this problem

3. The attempt at a solution

I have shown that

$$S_N(x)$$

can be written as

$$S_N(x)=\frac{2}{\pi} \int_0^{x} \frac{\sin (2 N t)}{\sin (t) } \; d{t}$$

my guess for suitable small angle formula is

$$\sin (x) \approx x$$ when x is small

Thank you for any help

Last edited: Mar 2, 2009
2. Mar 3, 2009

ghostyc

anyone got any ideas? :P