Infinite Sequences & Series

Visualize geometric convergence and discrete summing bounds. Slice the unit block, animate Zeno's infinite runner, and plot partial sums.

Zeno's Slices

Slicing & Runner Sandbox

An infinite series summates a sequence of numbers forever. In Zeno's paradox, a runner goes $1/2$ the remaining distance at each step. This geometric summation packing area always bounds to exactly 1.

Current Series State

CONVERGES
S_10 = 0.999
Starting Term (a) 0.50
Common Ratio (r) 0.50
Number of Terms (n) 10
Analytical Sum Estimation:
Sum = a / (1 - r) = 0.50 / 0.50 = 1.000
S_10 = 0.999 (99.9% of limit reached)
Anatomy Specs

Symmetry & Convergence Rules

Not all infinite series add up to a finite number. Understanding criteria determines convergence.

Geometric Series Rules


Convergence Condition:
A geometric series converges **if and only if** the common ratio satisfies **|r| < 1**. Under this state, the terms approach zero exponentially.
Divergence:
If **|r| ≥ 1**, terms grow or alternate with static size, causing the cumulative summation to shoot to positive/negative infinity or alternate.
Sum Formula:
S_∞ = a / (1 - r)

Harmonic & Arithmetic Rules


Harmonic Series (Σ 1/n):
The harmonic terms (1, 1/2, 1/3, 1/4...) approach zero, yet the overall sum **always diverges** to infinity, albeit very slowly. This proves that terms going to zero is not a sufficient condition for convergence!
Arithmetic Series (Σ a + nd):
Unless starting value a = 0 and difference d = 0, arithmetic summations **always diverge** to infinity because terms do not approach zero.
Self Evaluation

Sequences & Series Quiz

Assess your mathematical grasp of sequence boundaries, geometric convergence ratios, and series properties.

Question 1 of 5 Score: 0/0

Under what condition does the infinite geometric series Σ a rⁿ converge to a finite sum?

Correct Answer!

Explanation goes here.

Focus Area Vocabulary


  • Convergence: Summation values flattening to approach a finite limit.
  • Divergence: Summation values growing without bound to infinity.
  • Partial Sum (S_n): The sum of the first n terms of a sequence.