`sum_(x=0)^n(r^x sin(x alpha)) = ( "r" sin( alpha ) - "r" ^( "n" +1) sin[( "n" +1) alpha ] + "r" ^( "n" +2) sin( "n" alpha ) ) / (1 - 2 "r" cos( alpha ) + "r" ^2)`
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This is the sum of an progression of integers (`r^n`) times the sine of a function of `n * alpha`: `r sin(alpha) + r^2 sin(2 alpha)+...+ r^n sin(n alpha)`
Reference
This formula is also found in Schaum's Mathematical Handbook (19.44)