Alternating series theorem: Difference between revisions

Revision as of 21:05, 5 September 2011

Consider a series of the form:

$a_{1}+a_{2}+\dots +a_{n}+\dots =\sum _{k=1}^{\infty }a_{k}$

Suppose the following three conditions hold for the series:

Alternating signs: All the $a_{k}$ s are nonzero and the sign of $a_{k+1}$ is opposite the sign of $a_{k}$ .
Monotonically decreasing in magnitude: $|a_{k}|\geq |a_{k+1}|$ for all $k$ .
Terms approach zero: $\lim _{k\to \infty }a_{k}=0$ .

Then the series is a convergent series. It may be an absolutely convergent series or a conditionally convergent series, depending on whether the series of the absolute values of its terms converges.

Revision as of 21:04, 5 September 2011 (view source) Vipul (talk \| contribs) (Created page with "==Statement== Consider a series of the form: <math>a_1 + a_2 + \dots + a_n + \dots = \sum_{k=1}^\infty a_k</math> Suppose the following three conditions hold for the serie...")		Revision as of 21:05, 5 September 2011 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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	# ''Terms approach zero'': <math>\lim_{k \to \infty} a_k = 0</math>.		# ''Terms approach zero'': <math>\lim_{k \to \infty} a_k = 0</math>.

	Then the series converges.		Then the series is a [[convergent series]]. It may be an [[absolutely convergent series]] or a [[conditionally convergent series]], depending on whether the series of the absolute values of its terms converges.