5.6 The ratio test measures tail shrink speed · Sequences and Series, Second Half · Visual Calculus

5.6Ratio test

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Chapter 2 · Applications of Integration

2.1-2.2Areas & Slicing

2.3-2.4Shells & Arc Length

2.5Physical Applications

2.6Centers of Mass

2.7-2.9Log, Exp & Hyperbolic

Chapter 3 · Techniques of Integration

3.1-3.4Techniques I

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

Chapter 7 · Parametric Equations and Polar Coordinates

7.1-7.2Parametric Curves

7.3-7.4Polar Curves

Chapter 6 · Power Series

6.1-6.2Power Series I

6.3-6.4Power Series II

Home Calculus IISections 5.4-5.65.6 The ratio test measures tail shrink speed

5.6

The ratio test measures tail shrink speed

case

nn-factorial10

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Live setup

SymPy check = 2.7182818285

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