Problem $04\mathrm{.}017-$ Use a known estimate of $\frac{1}{1-x}$ and investigate error based on terms.

If it is known that

$\frac{1}{1-x}=1+x+x^2+x^3+$cdots

Add terms one at a time to estimate $\frac{1}{1-x}.$ Use a starting value of $x=0\mathrm{.}1.$ After each new term is added, compute the true and approximate percent relative errors. Use your calculator or Python to determine the true value. Add terms until the absolute value of the approximate error estimate falls below an error criterion corresponding to three significant figures. Use the values obtained to complete the following table: