A simple problem

Calculus Level 4

Denote f(x)=n=1Tnxn \displaystyle f(x) = \sum _{ n=1 }^{ \infty }{ { T }_{ n }{ x }^{ n } } where domain of f(x)f(x) are all values of xx where the summation converges, and Tn{T}_{n} is the nthn^{\text{th}} term of some sequence.

We define Sn=T1+T2+....+Tn {S}_{n} = {T}_{1}+{T}_{2}+ .... +T_{n}

And g(x)=n=1Snxn \displaystyle g(x) = \sum _{ n=1 }^{ \infty }{ { S }_{ n }{ x }^{ n } } with domain of g(x)g(x) are all values where this summation converges.

Lastly, denote h(x)=f(x)g(x)h(x) = \dfrac { f(x) }{ g(x) } , domain of h(x)h(x) is the intersection of domain of f(x),g(x)f(x),g(x) and values where g(x)0g(x) \neq 0

Then find x+h(x) x+h(x) in its domain.

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