What is the sum of: $$6 + 66 + 666 + 6666 + \cdots + 666666666\cdots6$$ where there are $n$ sixes in the last term?
Many functions, including the trigonometric and exponential functions that you meet in school, can be approximated by infinite power series and good approximations can be found using a finite number of terms. If the series is centred at zero then it can be written in the form $\Sigma_{n=0}^\infty a_nx^n$ where the coefficients depend on the derivative of the function at the origin. The infinite geometric series $1 + x + x^2 + \cdots$ which converges for $|x| < 1$ is the power series for the function $(1 - x )^{-1}$.
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