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STEP Support Programme

III 2015 Q4 iii)

Maybe I'm reading it wrong, but how come S1, S2, S3 are real when S1 = z1 + z2 + z3 = r(cos(theta) + isin(theta)) + r^2(cos(2theta) + isin(2theta)) + .... ?

Thank you!

Can't really see what you're asking: just checked the question and there's no mention of $S1, S2$ and $S_3$ being real, at least not at the outset?

Sorry! Forgot to say that that's what it said in the mark scheme on the ATS website

They're saying that the $S_i$ are real if the $S_i$ obey the conditions imposed in part (iii) of the question. That is, those conditions on $\theta_k$ and $r_k$ imply that $\Im(S_i) = 0$; so they show that the $S_i$ are real. If there's a step of the proof you don't understand, point it out.

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