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Determine the radius of convergence of (1 – (-2)n)zn

Determine the radius of convergence r of the power series:

    \[ \sum_{n=1}^{\infty} (1 - (-2)^n)z^n. \]

Test for convergence at the boundary points if r is finite.


We have

    \begin{align*}  \sum_{n=1}^{\infty} \left| (1 - (-2)^n)z^n \right| &= \sum_{n=1}^{\infty} \left|2^{n+1} \right| \left|z^n\right| \\[9pt]   &= 2 \sum_{n=1}^{\infty} \left| 2z \right|^n. \end{align*}

This is a geometric series which converges if and only if |2z| < 1 which implies |z| < \frac{1}{2}. Hence, r = \frac{1}{2}, and this converges at none of the boundary points.

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