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 Nov 21 asked Show that the function $h(x) = \int^x_0 g(t) \, dt$ is $C^2$ but not $C^3$ at $x = 0$ Nov 13 revised Books for high school students starting on college math added 182 characters in body Nov 13 answered Books for high school students starting on college math Nov 1 awarded Benefactor Nov 1 accepted When - during the study and development of- - and how were complex numbers introduced in the study of [real-valued] power series (expansions)? Oct 28 awarded Promoter Oct 5 asked When - during the study and development of- - and how were complex numbers introduced in the study of [real-valued] power series (expansions)? Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ So, -to clarify this whole debate-, in the space of real numbers I cannot write an expression like $S$, and claim that I can recast it [as $1 + x^2\,S$] unless the expression is convergent, (or defined within a specified radius of convergence)? Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ ... of convergence.] Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ @t.b.: Within the radius of convergence the series is identified with / [or possibly mathematically equivalent to(?)] $\frac{1}{1-x^2}$. My question is: [preliminarily, and from receiving various responses: can we recast it the way we have if $S$ is not convergent?, and] (if we can recast it regardless or not the series is convergent or divergent [which I now believe the answer is a 'no' to]) how is it mathematically justified? .. [Part of my question I think has already been answered since I don't believe you can recast it as above if the series isn't convergent [or outside its domain... Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ If I understand correctly, the series in the question can be recast only if it is convergent; otherwise it can't. [via @Jyrki | ... so, how then are "formal power series" related to the "usual" series above? ... Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ @Jyrki: Ok. Thanks for the response. -- May I courteously request a formal answer to the above question? (Use whatever mathematics you like, as long as your answer is completely rigorous.) - cheers Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Hi @Mark, thanks for the response. | May I know the formal power series version of your response? (I think that's what I'm searching for.) Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ @Jyrki: This would only apply if the series converged. | However, for this series, the recasting is valid regardless of convergence or divergence of the series, (i.e. the issue revolves around the specific infinite number of terms for the series - which allows recasting). ... My question is: how (in a rigorous mathematical sense) is this mathematically meaningful / (possible)? Oct 5 comment Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Well, I'm basically trying to understand why we can do this recasting. Normally - (I'm assuming) - we cannot recast a finite series. But the above recasting is allowed since the terms go to infinity. ... What is the exact mathematical (i.e. rigorous) basis which allows this manipulation? Oct 5 comment General question on relation between infinite series and complex numbers Hi @anon, your answer seems well-thought-out and valuable to someone with complex analysis under their belt, but I'm afraid currently it's somewhat over my head. (I'm still giving it +1 since I will review it once I gain sufficient knowledge / background in the subject.) - cheers Oct 5 asked Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Sep 30 answered Advantages of IMO students in Mathematical Research Sep 30 revised General question on relation between infinite series and complex numbers added 253 characters in body Sep 30 comment General question on relation between infinite series and complex numbers Can you provide a summary which is comprehensible to a 2nd-3rd year undergrad with modest knowledge of real analysis, and almost none of complex analysis (apart from complex numbers and some basic identities)?