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Thank you for the prompt response!

This is a simplified version of my previous question.

1) Is it possible to define a formal power series in sage by giving an expression for the n-th coefficient, e.g. as the expression "n" defines the power series 0 + 1 x + 2 x^2 + 3 x^3 + ... n x^n + ... ?

2) Does sage know how to multiply such objects by convolving the terms? Can it anti-differentiate them symbolically?

I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. It gets pretty messy so I was hoping that sage could keep track of the details for me. Here are the things I don't know how to do:

1) Define a Laurent series by giving an expression for its n-th coefficient.

2) Formally multiply and integrate Laurent series, producing an explicit expression for the n-th coefficient of the result.

Is this possible? I apologize if some or all of this is explained elsewhere.

EDIT: An example of what I want to do would be to define a power series such as sum(n * x^n,n,0,infinity) and then integrate it and/or multiply it by another power series, resulting in yet another power series whose coefficients I can read off. The O(x^n) notation is not very useful to me since I would like to have an expression for an arbitrary coefficient of that resulting series in terms of n.