(Solution) - Use this Taylor series to show that for any 0 -(2025 Original AI-Free Solution)

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Use this Taylor series to show that

for any 0 < q < 1.
We found that the function f(x) = 1/1 - x is equal to the Taylor series
Tf(x) = l + x + x2 + x3 + x4 + ...
when 0 < x < 1 (Example 3.7.10).