[JEE Advanced 2003] prove there is no complex z with z 13 and sum r 1 n
Prove there is no complex $z$ with $|z|<\tfrac13$ and $\sum_{r=1}^{n}a_rz^r=1$ where $|a_r|<2$.
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✓ Vidaara Team
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· 4d ago
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Answer: Proved (no such $z$).
$\left|\sum a_rz^r\right|\le\sum|a_r||z|^r<2\sum_{r=1}^{\infty}\left(\tfrac13\right)^r=2\cdot\frac{1/3}{1-1/3}=1$. So the sum's modulus is $<1$ and can never equal $1$.
JEE Advanced 2003 · Complex Numbers — verified solution by the Vidaara Team.
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