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UTD / MATHEMATICAL SCIENCE / MATH 2414 / What is alternating series test theorem?

# What is alternating series test theorem? Description

##### Description: These notes cover chapter 11.5 of Integral Calculus namely Alternating Series and Ratio Test
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"}.lst-kix_5ospsnx0h9zr-5 > li{counter-increment:lst-ctn-kix_5ospsnx0h9zr-5}.lst-kix_kwnzq1u7wrrj-7 > li{counter-increment:lst-ctn-kix_kwnzq1u7wrrj-7}.lst-kix_5ospsnx0h9zr-4 > li{counter-increment:lst-ctn-kix_5ospsnx0h9zr-4}.lst-kix_kwnzq1u7wrrj-4 > li{counter-increment:lst-ctn-kix_kwnzq1u7wrrj-4}ol.lst-kix_apri2b90k828-6.start{counter-reset:lst-ctn-kix_apri2b90k828-6 0}.lst-kix_pihy4e5vwgum-0 > li{counter-increment:lst-ctn-kix_pihy4e5vwgum-0}.lst-kix_pihy4e5vwgum-3 > li{counter-increment:lst-ctn-kix_pihy4e5vwgum-3}ol.lst-kix_pihy4e5vwgum-5.start{counter-reset:lst-ctn-kix_pihy4e5vwgum-5 0}.lst-kix_2p5n5k9uoeyw-6 > li{counter-increment:lst-ctn-kix_2p5n5k9uoeyw-6}.lst-kix_2p5n5k9uoeyw-3 > li{counter-increment:lst-ctn-kix_2p5n5k9uoeyw-3}ol.lst-kix_kwnzq1u7wrrj-1.start{counter-reset:lst-ctn-kix_kwnzq1u7wrrj-1 0}ol.lst-kix_kwnzq1u7wrrj-7.start{counter-reset:lst-ctn-kix_kwnzq1u7wrrj-7 0}ol.lst-kix_pihy4e5vwgum-8.start{counter-reset:lst-ctn-kix_pihy4e5vwgum-8 0}.lst-kix_5ospsnx0h9zr-7 > li{counter-increment:lst-ctn-kix_5ospsnx0h9zr-7}.lst-kix_apri2b90k828-6 > li{counter-increment:lst-ctn-kix_apri2b90k828-6}.lst-kix_pihy4e5vwgum-4 > li{counter-increment:lst-ctn-kix_pihy4e5vwgum-4}ol.lst-kix_apri2b90k828-5.start{counter-reset:lst-ctn-kix_apri2b90k828-5 0}.lst-kix_apri2b90k828-0 > li{counter-increment:lst-ctn-kix_apri2b90k828-0}ol.lst-kix_5ospsnx0h9zr-4.start{counter-reset:lst-ctn-kix_5ospsnx0h9zr-4 0}ol.lst-kix_kwnzq1u7wrrj-2.start{counter-reset:lst-ctn-kix_kwnzq1u7wrrj-2 0}ol.lst-kix_kwnzq1u7wrrj-7{list-style-type:none}.lst-kix_kwnzq1u7wrrj-8 > li{counter-increment:lst-ctn-kix_kwnzq1u7wrrj-8}.lst-kix_apri2b90k828-7 > li{counter-increment:lst-ctn-kix_apri2b90k828-7}ol.lst-kix_kwnzq1u7wrrj-6{list-style-type:none}ol.lst-kix_kwnzq1u7wrrj-8{list-style-type:none}ol.lst-kix_kwnzq1u7wrrj-3{list-style-type:none}ol.lst-kix_kwnzq1u7wrrj-2{list-style-type:none}ol.lst-kix_pihy4e5vwgum-2.start{counter-reset:lst-ctn-kix_pihy4e5vwgum-2 0}ol.lst-kix_kwnzq1u7wrrj-5{list-style-type:none}ol.lst-kix_kwnzq1u7wrrj-4{list-style-type:none}ol.lst-kix_kwnzq1u7wrrj-1{list-style-type:none}ol.lst-kix_kwnzq1u7wrrj-0{list-style-type:none}ol.lst-kix_2p5n5k9uoeyw-4.start{counter-reset:lst-ctn-kix_2p5n5k9uoeyw-4 0}.lst-kix_kwnzq1u7wrrj-2 > li{counter-increment:lst-ctn-kix_kwnzq1u7wrrj-2}.lst-kix_apri2b90k828-1 > li{counter-increment:lst-ctn-kix_apri2b90k828-1}ol.lst-kix_5ospsnx0h9zr-3.start{counter-reset:lst-ctn-kix_5ospsnx0h9zr-3 0}.lst-kix_2p5n5k9uoeyw-8 > li{counter-increment:lst-ctn-kix_2p5n5k9uoeyw-8}.lst-kix_pihy4e5vwgum-8 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-8,lower-roman) ". "}.lst-kix_2p5n5k9uoeyw-5 > li{counter-increment:lst-ctn-kix_2p5n5k9uoeyw-5}.lst-kix_2p5n5k9uoeyw-2 > li{counter-increment:lst-ctn-kix_2p5n5k9uoeyw-2}.lst-kix_pihy4e5vwgum-5 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-5,lower-roman) ". "}.lst-kix_pihy4e5vwgum-6 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-6,decimal) ". "}ol.lst-kix_kwnzq1u7wrrj-3.start{counter-reset:lst-ctn-kix_kwnzq1u7wrrj-3 0}.lst-kix_pihy4e5vwgum-7 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-7,lower-latin) ". "}.lst-kix_5ospsnx0h9zr-0 > li{counter-increment:lst-ctn-kix_5ospsnx0h9zr-0}ol.lst-kix_2p5n5k9uoeyw-5{list-style-type:none}.lst-kix_2p5n5k9uoeyw-7 > li:before{content:"" counter(lst-ctn-kix_2p5n5k9uoeyw-7,lower-latin) ". "}ol.lst-kix_2p5n5k9uoeyw-4{list-style-type:none}.lst-kix_pihy4e5vwgum-1 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-1,lower-latin) ". "}.lst-kix_pihy4e5vwgum-2 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-2,lower-roman) ". "}ol.lst-kix_2p5n5k9uoeyw-3{list-style-type:none}.lst-kix_2p5n5k9uoeyw-6 > li:before{content:"" counter(lst-ctn-kix_2p5n5k9uoeyw-6,decimal) ". "}.lst-kix_2p5n5k9uoeyw-8 > li:before{content:"" counter(lst-ctn-kix_2p5n5k9uoeyw-8,lower-roman) ". "}.lst-kix_5ospsnx0h9zr-6 > li{counter-increment:lst-ctn-kix_5ospsnx0h9zr-6}ol.lst-kix_2p5n5k9uoeyw-2{list-style-type:none}ol.lst-kix_2p5n5k9uoeyw-8{list-style-type:none}.lst-kix_pihy4e5vwgum-0 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-0,decimal) ". "}.lst-kix_pihy4e5vwgum-4 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-4,lower-latin) ". "}.lst-kix_5ospsnx0h9zr-3 > li{counter-increment:lst-ctn-kix_5ospsnx0h9zr-3}ol.lst-kix_2p5n5k9uoeyw-7{list-style-type:none}ol.lst-kix_2p5n5k9uoeyw-5.start{counter-reset:lst-ctn-kix_2p5n5k9uoeyw-5 0}ol.lst-kix_2p5n5k9uoeyw-6{list-style-type:none}ol.lst-kix_2p5n5k9uoeyw-1{list-style-type:none}ol.lst-kix_apri2b90k828-4.start{counter-reset:lst-ctn-kix_apri2b90k828-4 0}ol.lst-kix_2p5n5k9uoeyw-0{list-style-type:none}.lst-kix_pihy4e5vwgum-3 > li:before{content:"" counter(lst-ctn-kix_pihy4e5vwgum-3,decimal) ". "}.lst-kix_5ospsnx0h9zr-0 > li:before{content:"" counter(lst-ctn-kix_5ospsnx0h9zr-0,decimal) ". "}ol.lst-kix_apri2b90k828-8{list-style-type:none}.lst-kix_pihy4e5vwgum-7 > li{counter-increment:lst-ctn-kix_pihy4e5vwgum-7}ol.lst-kix_apri2b90k828-7{list-style-type:none}ol.lst-kix_pihy4e5vwgum-3.start{counter-reset:lst-ctn-kix_pihy4e5vwgum-3 0}

11.5 Alternating Series

1. Theorem: alternating series test

If the alternating series (-1)n-1bn = b1- b2 + b3 - b4 - b5 - b6 + … > 0If you want to learn more check out What is a direct comparison test?

SatisfiesDon't forget about the age old question of What does trait theories focuses on?

A1 bn+1 < bn Vn (decreasing) or < 1We also discuss several other topics like The actual story of the banjo begins when?

b1 bn = 0

Then the series convergesDon't forget about the age old question of The medial amygdala is in charge of?

Example:

(-1)n-1

<;= 0If you want to learn more check out What is the amount of the discount on these bonds at issuance?

∴ the series converges

b1(-1)n+1ne-nDon't forget about the age old question of Who is Anne Bogart?

Bn = ne-n  ==.=< 1

= 0

∴ the series converges

C1(-1)n-1 arctan(n): arctan is an increasing function

arctan(n) 0

(-1)n-1 arctan(n) = DNE 0

∴ the series diverges by divergence test

d1(-1)n-

1. Estimating Sums

Remainder: Rn = S - Sn

Theorem: alternating series estimation

If S = (-1)n+1bn is the sum of a series that satisfies;

a1 bn+1 = bn

b1bn = 0

Then |Rn| = |S - Sn| < bn+1

Sn < S < Sn+1

Example:

Rn < bn+1 = == .000064 < 1

11.6 Absolute Convergence, Ratio and Root Test

1. A series an is called absolutely convergent if the series |an| is convergent

Example: (-1)n+1| = converges by p - series

Example: =< 1 ∴decreasing = 0

converges., but not absolutely

1. Definition: A series an is called conditionally convergent if it is convergent but not absolutely convergent
2. Theorem: if the series an is absolutely convergent, it’s converge

Proof: an

an+ |an| < 2|an|  (an > 0  → an + |an| = 0 an = 0 → an + |an| = 29n)

The series an is absolutely convergent means |an|

→ 2 |an| = 2|an| converges

an converges by comparison test

Example: Determine if series convergence / divergence

a1

|| → <

is absolutely convergent by direct comparison test

b1

|r| < 1, the series converges by absolute convergence test

c1bnn =         =< 1 decreasing

= 0

is conditionally convergent

= DNE

Ratio Test

Theorem: Let an be a series with nonzero terms

a1an converges absolutely if < 1

b1an diverges if > 1

C1 the test is inconclusive if  = 1

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