Question
Find the radius of convergence of the following series.
i)
ii)
Answer :
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For a power series (\sum a_k (z-z_0)^k), the radius of convergence (R) is given by [ \frac{1}{R}=\limsup_{k\to\infty}|a_k|^{1/k}. ] i) The given series is [ \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k!}(z-(1+i))^k. ] Here (a_k=\frac{(-1)^{k+1}}{k!}). Then [ |a_k|^{1/k}=\frac{1}{(k!)^{1/k}}. ] Using the fact
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