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Say if I define a power series over some arbitrary field $F$ as

$$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$

Then can I say:

$$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i + j} $$

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    $i_1+j_1=i_2+j_2$ for many pairs. just use the cauchy product (gathering up terms with the same exponent)2011-03-28
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    @user8603, @Bill Dubuque, @Mitch, @Jonas Kibelbek: My apologies for giving a wrong answer ("very misleading" was putting it mildly :-) -- I've deleted it. I wasn't aware of the notion of convergence for formal power series defined in Bill's answer -- thanks to all of you for educating me. Bill, your point about the "widespread confusion" about this is confirmed by the 7 upvotes I got for my "complete nonsense". :-)2011-03-28

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