Paper 1, Section II, D

Analysis I | Part IA, 2013

(a) Determine the radius of convergence of each of the following power series:

n1xnn!,n1n!xn,n1(n!)2xn2\sum_{n \geqslant 1} \frac{x^{n}}{n !}, \quad \sum_{n \geqslant 1} n ! x^{n}, \quad \sum_{n \geqslant 1}(n !)^{2} x^{n^{2}}

(b) State Taylor's theorem.

Show that

(1+x)1/2=1+n1cnxn(1+x)^{1 / 2}=1+\sum_{n \geqslant 1} c_{n} x^{n}

for all x(0,1)x \in(0,1), where

cn=12(121)(12n+1)n!c_{n}=\frac{\frac{1}{2}\left(\frac{1}{2}-1\right) \ldots\left(\frac{1}{2}-n+1\right)}{n !}

Typos? Please submit corrections to this page on GitHub.