Evaluate $360 (\frac{1}{15}+\frac{1}{105}+\frac{1}{315}+⋯).$ Evaluate $360$ ($\frac{1}{15}$+$\frac{1}{105}$+$\frac{1}{315}$+$\cdots$).
My Work :-
Well we can clearly see that $15 = 1*3*5$ ; $105 = 3*5*7$ ; $315 = 5*7*9$. So I basically know the pattern, but I want to know how to apply that like maybe we can break $\frac{1}{15}$ into $x*\frac{1}{1}\pm y*\frac{1}{3}\pm z*\frac{1}{5} $ or something like that so that we can cancel some terms in the whole sum.

Hint: $$\frac{1}{105}=\frac{1}{4}(\frac{1}{3\cdot 5}-\frac{1}{5\cdot 7})$$ $$\frac{1}{315}=\frac{1}{4}(\frac{1}{5\cdot 7}-\frac{1}{ 7\cdot 9})$$  do you
observe a pattern........
