An Introduction to Tauberian Theorems

An Introduction to Tauberian Theorems

Thu, 18/06/2009
16:00 		Timothy Trudgian (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
An Introduction to Tauberian Theorems
Suppose a power series $ f(x):= \sum_{n=0}^{\infty} a_{n} x^{n} $ has radius of convergence equal to $ 1 $ and that $ lim_{x\rightarrow 1}f(x) = s $. Does it therefore follow that $ \sum_{n=0}^{\infty} a_{n} = s $? Tauber's Theorem answers in the affirmative, if one imposes a certain growth condition (a Tauberian Condition) on the coefficients $ a_{n} $. Without such a condition it is clear that this cannot be true in general - take, for example, $ f(x) = \sum_{n=0}^{\infty} (-1)^{n} x^{n}. $