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QUESTION


Find the radius of convergence of the power series
$$\sum_{n=1}^\infty{z^n\over \sqrt n}.$$ Find one point on the
circle of convergence where this series converges and one point on
the circle of convergence where it diverges.



ANSWER


$R=\lim_{n\rightarrow\infty}{\sqrt n\over \sqrt{n+1}}$. At $z=1$
we have $1+{1\over \sqrt 2}+{1\over \sqrt 3}+\cdots$ which
diverges by the comparison test. At $z=-1$ the series converges by
the alternating series test.




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