06 January 2009

Pólya and Szegö take a dim view of authority

George Pólya and Gabor Szegö, in Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions, define the "maximum term" of a power series with all coefficients pi positive,

p0 + p1 x + p2 x2 + ...,

as that term for which pi xi is largest. (Obviously this depends on x.) They then ask:
[Part 1, Problem] 120. [Prove that] the subscript of the maximum term increases as x increases. (One might consider this situation as somewhat unusual: in the course of successive changes the position of maximum importance is held by more and more capable individuals.)
Problem 119 is to prove that for an everywhere convergent power series which is not a polynomial, the subscript of the maximal term goes to ∞ with x. Pólya and Szegö offer no commentary.


Ben Allen said...

So x^17 is more capable than x^4? I wouldn't trust either of them to do much more than deliver me large numbers.

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