> [!NOTE] Definition (Convergent series of reals) > Given a [[Real sequences|sequence]] of reals $(a_{n})$ is we say its [[Series of Real Sequence|series]] = A:$\sum_{n=1}^{\infty} a_{n} =A$If the sequence $(s_{k})$ of *partial sums* $s_{k} = \sum_{n=1}^{k}a_{n}$[[Convergence|converges]] to $A$ as $k \to \infty$. > [!NOTE] Notation > - $\sum_{n}^{\infty} a_{n} < \infty$ if the series converges, > - $\sum_{n}^{\infty} a_{n} = \infty$ if does not converge (diverges). # Properties Algebra of convergent series (these follow from algebra of limit on the sequence of partial sums): - [[Sum of Two Convergent Series is Convergent]]; - [[Constant Multiple of Convergent Series is Convergent]]; - [[Merten's Convergence Theorem]]. Criteria for convergence: - Note that the [[Terms of Convergent Series Tend to Zero]] follows from shift rule on sequence of partial sums. - [[Series with Non-Negative Terms Converges Iff Partial Sums Are Bounded Above]]. - [[Absolutely Convergent Series is Convergent]]. - See [[Tests for Convergence of Series]]. # Examples - [[Convergent Geometric Series]]. - [[Telescoping Sum|Telescoping sequences]]. - [[Harmonic Numbers]]. - [[Basel Problem]]. - [[p-series converges when p > 1]]. - [[Euler's Number]]. - [[Power Series]].