x+x^2+x^3+...x^n

x+x^2+x^3+...x^n → x(x^n - 1)/(x - 1)

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CPhill: What is the difference between this sequence and the question?

∑[x^(n), n, 1, 10] = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10.........etc.

this equals x/(1-x) for all -1

I think you meant to put the condition -1 < x < 1 Max; but notice your statement is only true if n is infinite.

Yes...

I have been less careful these days...