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## Definition: Real Infinite Series, Partial Sums

Let $$(x_n)_{n\in\mathbb N}$$ be a real sequence. The real sequence $$(s_n)_{n\in\mathbb N}$$ of partial sums
$s_n:=\sum_{k=0}^n x_k,\quad\quad n\in\mathbb N$
is called the (infinite) real series
$\sum_{k=0}^\infty x_k\quad\quad( * ).$

Note: If the sequence of partial sums is convergent, the expression $$( * )$$ also denotes the limit to which the sequence converges.

| | | | | created: 2015-02-18 14:20:32 | modified: 2016-12-22 23:09:59 | by: bookofproofs | references: [581]

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