Let \((a_n)\) be a sequence in a topological space . A series is convergent if the sequence of partial sums \((\sum_{n=0}^{N} a_n)_N\) is convergent . The limit is then denoted by \(\sum_{n=0}^{\infty} a_n\).
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Remarks
- If the terms are all positive real numbers, convergence is equal to absolute convergence .
- On normed spaces it is necessary that \((a_n)\) converges to zero.