Product of two power series

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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Yes, using the natural notion of convergence for formal power series, the stated sum does indeed converge to the Cauchy product. One should beware – as exemplified in this thread – that there is widespread confusion about formal vs. functional power series – even by some experts (in other fields). Rota frequently told jokes in his lectures about certain distinguished mathematicians who published complete nonsense based on such confusion (Indiscrete Thoughts!)

In any case the basic ideas are quite simple if you merely take off your analyst hat and, instead, put on your algebraist or combinatorist hat. In particular, you should be able to find a correct discussion of convergence of formal power series in almost any good book on combinatorics or generating functions, e.g. here is an excerpt from Stanley’s classic $\: $ Enumerative Combinatorics I.

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Yes, you can: this is the definition of the product of the two formal power series $a$ and $b =\sum_{i=0}^\infty b_i X^i$.

Not if you are talking about multiplying $a$ by a similarly defined formal power series $b$. The multiplication of formal power series can be written as:
$ab = \sum_{i=0}^{\infty}(\sum_{j=0}^ia_jb_{i-j})X^i$