Problem 1554 (difficulty: 3/10)

Find the radius of convergence of the following series. At which points do they converge, do they converge absolutely? What is their termwise derivative, antiderivative and what is the radius of convergence of those series? What is the largest disc with the same center as the power series to which these functions extend as regular functions?

\(\displaystyle \sum_{n=0}^\infty z^n; \qquad \sum_{n=0}^\infty (n+1)(z+1)^n \qquad \sum_{n=0}^\infty \frac{(z-i)^n}{n!}; \qquad \sum_{n=1}^\infty \frac{(z+i)^n}n. \)

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