MHB What Properties of Fourier Series Are Revealed by This Equation Transformation?

[nacho-man]

Is there some properties I should be aware of?

after making the relevant substitutions, I ended up with

$2 = 1 + \sum\nolimits_{m=0}^\infty \frac{4}{(2m+1)\pi}\sin(\frac{(2m+1)\pi}{2})$

but I can't get past this

 

[zzephod]

Is there some properties I should be aware of?

after making the relevant substitutions, I ended up with

$2 = 1 + \sum\nolimits_{m=0}^\infty \frac{4}{(2m+1)\pi}\sin(\frac{(2m+1)\pi}{2})$

but I can't get past this

Rearrange to:

$$\frac{\pi}{4}=\sum_{m=0}^\infty \frac{1}{2m+1}\sin\left(\frac{(2m+1)\pi}{2}\right)$$

and ask yourself what values does the sine of odd half multiples of $$\pi $$ take?

.