The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Geometric progressions: finite sums, convergence and remainders
Recover all possible ratios, including negative ones.
Distinguish finite sums from a convergent infinite sum.
Handle exceptional ratios and use the absolute omitted tail for error bounds.
Use \(u_n=ar^{n-1}\) and \(\displaystyle S_n=\frac{a(1-r^n)}{1-r}\) when applicable; for \(r=1\), \(S_n=na\). If \(a\ne0\), an infinite sum exists exactly when \(|r|<1\), and \(\displaystyle S_\infty=\frac a{1-r}\). The signed remainder is \(R_n=S_\infty-S_n\); an error tolerance uses \(|R_n|\). For \(r=0\), write \(a,0,0,\ldots\) directly.