Paper 1, Section I, F

Analysis I | Part IA, 2019

Let n=1anxn\sum_{n=1}^{\infty} a_{n} x^{n} be a real power series that diverges for at least one value of xx. Show that there exists a non-negative real number RR such that n=1anxn\sum_{n=1}^{\infty} a_{n} x^{n} converges absolutely whenever x<R|x|<R and diverges whenever x>R|x|>R.

Find, with justification, such a number RR for each of the following real power series:

(i) n=1xn3n\sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}};

(ii) n=1xn(1+1n)n\sum_{n=1}^{\infty} x^{n}\left(1+\frac{1}{n}\right)^{n}.

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