Define what it means for a series \(\sum_{n=1}^\infty a_n\) (of real numbers \(a_n\)) to converge to a real number \(S\). Prove that the series \(\sum_{n=1}^\infty 1/10^n\) converges.
Solution
1. The series \(\sum_{n=1}^{\infty} a_n\) converges to \(S\) if the sequence of partial sums \(s_N = \sum_{n=1}^{N} a_n\) converges to \(S\), i.e. \(\lim_{N \to \infty} s_N = S\).
definition of series convergence.
2. For \(a_n = 1/10^n\), the partial sums are \(s_N = \sum_{n=1}^{N} 10^{-n} = \frac{1/10 (1 - 10^{-N})}{1 - 1/10} = \frac{1}{9}(1 - 10^{-N})\).
geometric series formula.
3. \(\lim_{N \to \infty} s_N = \frac{1}{9}(1 - 0) = \frac{1}{9}\).
2 and \(10^{-N} \to 0\).
4. Hence \(\sum_{n=1}^{\infty} 1/10^n\) converges, to \(\frac{1}{9}\).
1 and 3.
5. Q.E.D.
4.