Analysis | Part IA, 2003

Explain what is meant by the radius of convergence of a power series.

Find the radius of convergence RR of each of the following power series: (i) n=1n2zn\sum_{n=1}^{\infty} n^{-2} z^{n}, (ii) n=1(n+12n)zn\sum_{n=1}^{\infty}\left(n+\frac{1}{2^{n}}\right) z^{n}.

In each case, determine whether the series converges on the circle z=R|z|=R.

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