5.4 A finite ratio transfers the verdict · Sequences and Series, Second Half · Visual Calculus

5.4Limit comparison

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Chapter 2 · Applications of Integration

2.1-2.2Areas & Slicing

2.3-2.4Shells & Arc Length

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2.6Centers of Mass

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3.5-3.7Techniques II

Chapter 5 · Sequences and Series

5.1-5.3Sequences & Series I

5.4-5.6Sequences & Series II

5.4Direct comparison 5.4Limit comparison 5.4Choosing comparators 5.5Alternating partial sums 5.5Alternating series test 5.5Alternating remainder 5.5Absolute vs conditional 5.5Rearrangement 5.6Ratio test 5.6Root test 5.6When rho = 1 5.6Choosing a test 5.6Pi series race

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Home Calculus IISections 5.4-5.65.4 A finite ratio transfers the verdict

5.4

A finite ratio transfers the verdict

case

root12

Interactive view

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Live setup

SymPy check = 1

SymPy checks

Geometric comparison bound1

Limit-comparison ratio1

Dominant geometric comparator sum2

Alternating harmonic sum0.6931471806

Alternating p=2 sum0.8224670334

Alternating p=2 N=10 error bound0.0082644628

Absolute-series harmonic comparison limit0.3333333333

OpenStax rearranged alternating harmonic sum1.0397207708

Ratio-test rho2.7182818285

Root-test rho0.25

p-series ratio and root boundary rho1

Example 5.25a limit-comparison ratio1

Series target pi3.1415926536