Alternating Series, Absolute Convergence and Conditional ..., Study Guides, Projects, Research of Advanced Calculus

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Alternating Series, Absolute

Convergence and Conditional

Convergence

n= 1

In an Alternating Series, every other term has the opposite sign. AST (Alternating Series Test) Let a 1 - a 2 + a 3 - a 4 +... be an alternating series such that an>an+ 1 > 0 , then the series converges. The error made by estimating the sum, Sn is less than or equal to an+ 1 , i.e. E = |S - Sn| ≤ an+ 1.

Absolute Convergence If converges, then converges. EX 3 Does converge or diverge?

Absolute Ratio Test Let be a series of nonzero terms and suppose. i) if ρ< 1 , the series converges absolutely. ii) if ρ > 1 , the series diverges. iii) if ρ = 1 , then the test is inconclusive. EX 4 Show converges absolutely.

Rearrangement Theorem The terms of an absolutely convergent series can be rearranged without affecting either the convergence or the sum of the series.