Editing Formal power series (section)

==== Alternative topologies ====
The above topology is the [[Comparison of topologies|finest topology]] for which

:<math>\sum_{i=0}^\infty a_i X^i</math>

always converges as a summation to the formal power series designated by the same expression, and it often suffices to give a meaning to infinite sums and products, or other kinds of limits that one wishes to use to designate particular formal power series. It can however happen occasionally that one wishes to use a coarser topology, so that certain expressions become convergent that would otherwise diverge. This applies in particular when the base ring <math>R</math> already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.

In the ring of formal power series <math>\Z[[X]][[Y]]</math>, the topology of above construction only relates to the indeterminate <math>Y</math>, since the topology that was put on <math>\Z[[X]]</math> has been replaced by the discrete topology when defining the topology of the whole ring. So

:<math>\sum_{i = 0}^\infty XY^i</math>

converges (and its sum can be written as <math>\tfrac{X}{1-Y}</math>); however

:<math>\sum_{i = 0}^\infty X^i Y</math>

would be considered to be divergent, since every term affects the coefficient of <math>Y</math>. This asymmetry disappears if the power series ring in <math>Y</math> is given the product topology where each copy of <math>\Z[[X]]</math> is given its topology as a ring of formal power series rather than the discrete topology. With this topology, a sequence of elements of <math>\Z[[X]][[Y]]</math> converges if the coefficient of each power of <math>Y</math> converges to a formal power series in <math>X</math>, a weaker condition than stabilizing entirely.  For instance, with this topology, in the second example given above, the coefficient of <math>Y</math>converges to <math>\tfrac{1}{1-X}</math>, so the whole summation converges to <math>\tfrac{Y}{1-X}</math>.

This way of defining the topology is in fact the standard one for repeated constructions of rings of formal power series, and gives the same topology as one would get by taking formal power series in all indeterminates at once. In the above example that would mean constructing <math>\Z[[X,Y]]</math> and here a sequence converges if and only if the coefficient of every monomial <math>X^iY^j</math> stabilizes. This topology, which is also the <math>I</math>-adic topology, where <math>I=(X,Y)</math> is the ideal generated by <math>X</math> and <math>Y</math>, still enjoys the property that a summation converges if and only if its terms tend to 0.

The same principle could be used to make other divergent limits converge. For instance in <math>\R[[X]]</math> the limit

:<math>\lim_{n\to\infty}\left(1+\frac{X}{n}\right)^{\!n}</math>

does not exist, so in particular it does not converge to

:<math>\exp(X) = \sum_{n\in\N}\frac{X^n}{n!}.</math>

This is because for <math>i\geq 2</math> the coefficient <math>\tbinom{n}{i}/n^i</math> of <math>X^i</math> does not stabilize as <math>n\to \infty</math>. It does however converge in the usual topology of <math>\R</math>, and in fact to the coefficient <math>\tfrac{1}{i!}</math> of <math>\exp(X)</math>. Therefore, if one would give <math>\R[[X]]</math> the product topology of <math>\R^\N</math> where the topology of <math>\R</math> is the usual topology rather than the discrete one, then the above limit would converge to <math>\exp(X)</math>. This more permissive approach is not however the standard when considering formal power series, as it would lead to convergence considerations that are as subtle as they are in [[analysis (mathematics)|analysis]], while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be. With this topology it would ''not'' be the case that a summation converges if and only if its terms tend to 0.