Convergent Series and Limit Theorems

Basic operations with convergent series.

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Concept Content
Limit Theorems Combining limits is a useful tool to deduce convergence and the limit of a more complicated sequence from the convergence of simpler building blocks.
Series and Partial Sums A series is a sequence of partial sums.
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Concept Content
Leibniz Criterion A convergence criterion for sums based on an alternating sequence.
Reordering Series can be reordered.
Integral Comparison Test Compare improper Riemann integrals to infinite series.
Comparison Test If a series converges can be checked with different tests.

Study Convergent Series and Limit Theorems

Next we consider sums of convergent series and multiplication of series by some scalar variable. The proof just consists of a straightforward application of the corresponding Theorems for sequences and is therefore skipped.

Theorem 1 (Formulae for convergent series). Let \(\lambda\in\mathbb{K}\) and \[\sum_{k=1}^\infty a_k,\qquad \sum_{k=1}^\infty b_k\] be two convergent series in \(\mathbb{K}\). Then

  1. \[\sum_{k=1}^\infty(a_k+b_k)=\sum_{k=1}^\infty a_k+\sum_{k=1}^\infty b_k;\]

  2. \[\sum_{k=1}^\infty(\lambda a_k)=\lambda \sum_{k=1}^\infty a_k.\]

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