By Morse M., Littauer S. B.
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Extra resources for A Characterization of Fields in the Calculus of Variations
On the other hand, for all those x where the power series converges, the value of the sum S(x) == L~o aix' defines a value of a real function S. Power series thus provide a natural bridge leading us to the consideration of more general real functions. e. those whose definition 'transcends algebra' since they cannot be built as ratios or other algebraic combinations of polynomials, are nevertheless limits of polynomials in this natural way. We now examine the power series leading to such functions in more detail, concentrating on the well-known examples of the exponential and trigonometric functions, that is, we shall justify the definitions: n exp(x) == 3 n=O 00 sin(x) 2 oox x x" X x L, n.
This set is either bounded or unbounded above. We consider both cases separately. If C is bounded above, then it contains only finitely many elements of N. If C is empty, let nl = 1. Otherwise we can find nl E N greater than the maximum of C, so none of the numbers nl, ni + 1, nl + 2, ... belong to C. But if ns is not in C there must exist nz E N such that ni > nl and X n 2 ~ x n 1 • Now nz is greater than nl, so it cannot belong to C either. } constructed in this way is increasing. On the other hand, if C is unbounded above, we can find an infinite sequence of natural numbers nl < n2 < n3 < ...
He then drew in vertical lines to describe the area of this 'infinite tower'. What happens when the tower collapses? I I -I-- I I I I I : I 11 II II 1 I 1 1 2" 4 8 Fig. 1 I I I D Oresme's Tower I In finding limits of sequences we have so far had to 'guess' the limit first, and then gone on to prove that our guess was correct. It would be much more satisfactory if we could find a recipe for deducing that certain types of sequences will always converge. By the same token it will be useful later to have a series of tests which can be used to decide when a given series converges - even if the tests will usually not be able to give us much information on the value of the sum of the series.
A Characterization of Fields in the Calculus of Variations by Morse M., Littauer S. B.