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I read Wikipedia article about formal power series, but I don't get an intuitive idea... I had several doubts regarding this topic

1)Why we need formal power series

2)Formal power series does not give any significance to convergence, then why we used a diverging mathematical entity?

3)I read formal power series as a set of coefficients, I don't get what it means?

4)Can we represent a converging function($1+x$) by a formal power series?

5)Can we represent a function with a pole ($1\over{1+x}$) for $x>>1\&x<<1$ by a formal power series?

Please help me to get an intuitive idea rather than some mathematical formulae

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    $\begingroup$ They present an interesting algebraic structure. You can add and multiply them the way you would polynomials. To some extent one can perform the operations of calculus on them without worrying about limits or anything like that. $\endgroup$
    – lulu
    Commented Aug 25, 2020 at 14:53
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    $\begingroup$ Wilf described a generating function (a formal power series) as "a clothesline on which we hang up a sequence of numbers for display" math.upenn.edu/~wilf/gfology2.pdf $\endgroup$ Commented Aug 25, 2020 at 15:51

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It extends the concept of polynomial to an infinity of terms. And it has nothing to do with convergence or divergence, since it is not a sum of functions. You're confusing formal power series and functions defined by power series. Formal power series are defined for any commutative ring of coefficients.

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A formal power series is indeed formally equivalent to a sequence of numbers.

$$a_0,a_1,a_2,\cdots\leftrightarrow a_0+a_1x+a_2x^2+\cdots$$

because you easily turn one into the other.

They are useful in that

  • they coincide with an entire series where the latter converges,

  • they are compatible with the logics for adding or multiplying polynomials and entire series (term-wise addition and convolutions),

  • they can convey non-converging generating functions and draw combinatorial properties.

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  • $\begingroup$ "they coincide with an entire series where the latter converges," Is that statement your opinion or is it a mathematical theorem? I'm also curious about that. You're expressing this idea, right? $\endgroup$ Commented Sep 16, 2023 at 15:18
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One application of generating functions is to solve counting problems like with Rook Polynomials. In this setting the coefficients will count a slight variant of the general problem and there are often infinitely many. This is useful because we can often calculate the coefficients in isolation without having to calculate the other terms to solve these problems.

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