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Introduction to Power Series
15. Power Series / Introduction to Power Series / Problem 3
Problem 3

Consider the power series defined by f(x)=∑k=0∞xk=11−x\(\displaystyle\) f(x) = \(\sum\)_{k=0}^{\(\infty\)} x^k = \(\frac{1}{1-x}\), for ∣x∣<1|x| < 1, and the partial sum of the first nn terms given by Sn(x)=∑k=0n−1xk{\(\displaystyle\) S_{n}(x)=\(\sum\)_{k=0}^{n-1}x^{k}}. Define the remainder after nn terms as Rn(x)=f(x)−Sn(x)R_{n}(x)=f(x)-S_{n}(x). Can the remainder be expressed as Rn(x)=xn1−xR_{n}(x)=\(\frac{x^n}{1-x}\)?