Series
is the series with terms , whenever the limit of partial sums exists.
Definition 5.26 (Series)
Let be a sequence in . Then
is the th partial sum of the series whose th term is . We write
if the limit exists.
Examples
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The geometric series with ratio has partial sums , so .
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The harmonic series diverges: grouping the terms shows , so the partial sums are increasing and unbounded.
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The series with terms converges: its partial sums are increasing and bounded above by , so the Monotonic Convergence Theorem applies; in fact .
Related
Stated in
- Definition 5.26 (Series)§5.3 Series