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VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 4 Theory

Infinite Sequences & Series Convergence Tests

Ratio, Root, Integral, and Alternating Series bounds

Plot discrete partial sums S_N against continuous integral bounds to understand why the Ratio Test measures geometric shrinkage.

1. The Ratio Test & Geometric Domination

If the ratio of consecutive terms |a_{n+1} / a_n| approaches a limit L < 1, the series eventually shrinks faster than a convergent geometric series with ratio r = (L + 1)/2.

L=lim⁡n→∞∣an+1an∣{<1Absolute Convergence>1Divergence=1InconclusiveL = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \begin{cases} < 1 & \text{Absolute Convergence} \\ > 1 & \text{Divergence} \\ = 1 & \text{Inconclusive} \end{cases}

Quick Reference & Cheat Sheet

Alternating Series Error Bound
∣S−SN∣≤bN+1|S - S_N| \le b_{N+1}
The truncation error is strictly smaller than the first omitted term.