A <convergent series> $\sum_{n\geq1}u_n$ has <partial sums> $S_N=\sum_{n=1}^Nu_n$ tending to a finite limit $S$. Explicitly, for every $\varepsilon>0$ there is an integer $N_0$ such that $|S_N-S|<\varepsilon$ whenever $N\geq N_0$. The convergence of the individual terms to zero is necessary but is not sufficient for convergence of the series.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 10 / a / Solution

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