5.1 An epsilon band makes convergence visible · Sequences and Series, First Half · Visual Calculus

5.1Epsilon band

∫Visual Calculus

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

5.1Epsilon band 5.1Geometric sequences 5.1Squeeze theorem 5.1Monotone + bounded 5.1Cobweb recurrence 5.2Partial sums 5.2Geometric series 5.2Telescoping series 5.2Harmonic series 5.3Divergence test 5.3Integral test 5.3p-series 5.3Remainder estimate

5.4-5.6Sequences & Series II

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 5.1-5.35.1 An epsilon band makes convergence visible

5.1

An epsilon band makes convergence visible

sequence

monotone0.2

Interactive view

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Live setup

SymPy check = 1

SymPy checks

Limit of epsilon-band sequence1

Limit of geometric sequence0

Squeezed sequence limit0

Monotone bounded sequence limit0

Positive fixed point of recurrence1

Oil-seepage geometric sum2

Alternating geometric sum5.1428571429

Telescoping series sum1

Convergent p-series contrast1.6449340668

Divergence-test term limit0.3333333333

Integral-test p=2 tail1

p-series tail integral at p=40.3333333333

Remainder upper bound at N=100.005