Power series convergence
What are some general techniques for showing that a power series expansion of a function converges to that function? As an important example, if you’re dealing with a real-valued function, what techniques exist for showing it is real-analytic, other than dealing directly with epsilon-delta limit arguments?
The only power series equality that I can remember proving directly is 
when 
. In real analysis, I can't remember ever proving a function is real-analytic (not that we didn't ever do it, maybe we did), and in complex analysis, we use Cauchy-Reimann to show that a function is analytic, so equal to its power series.
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- Regular Series (7/20/2007)
- Auditing a Fourier course (6/9/2005)
- Two ways of doing cool stuff with matrices (2/27/2007)