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Talk:Formal power series: Difference between revisions
→Multiplication and division: What is the task about?
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:the solution is infinite. And we ignored the convergence issue, so far. --[[User:Dmitry-kazakov|Dmitry-kazakov]] 09:36, 11 March 2009 (UTC)
:: There are several real problems. One is that we can't expand in 0 if the function (or any of the derivative) diverges in 0. Instead of sin, use the 1/x function itself to find its coefficients "directly" the same way you use for sin... Another problem is that we expand in powers of x where the exponent is positive, so it exists no expansion for x<sup>-1</sup>. We should drop Taylor series and use Laurent series (always with care I suppose). If we can "generate" the Taylor series in 0 for r(x), being r(x) = f(x)/g(x), then we can divide the expansion of f(x) (its formal power series) by the one for g(x). There are still problems anyway; e.g. if it exists N so that f<sub>i</sub> = 0 for i>N, and similar for g<sub>i</sub> but with M, and N<M, what happens? (Similar to 1/x case I suppose), at a glance this situation gives problem too. But this still does not say that division is '''always''' impossible. It is "sometimes"! --[[User:ShinTakezou|ShinTakezou]] 19:15, 11 March 2009 (UTC)
:::Yep, the task name is misleading. When it says "formal series" that gives an impression of some generalized framework for dealing with more or less '''any''' series, Fourier, Chebyshev, not just Taylor ones. Laurent series is yet another story. It goes in direction of approximations by rational polynomials, Padé etc. So what is the task about? The example of a definition of cos-sin has almost nothing to do with either series or approximations and how they are dealt with in real world. (I addressed this issue in a subthread.) --[[User:Dmitry-kazakov|Dmitry-kazakov]] 09:44, 12 March 2009 (UTC)
==About solving equations==
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