Each term is a constant multiple of the previous term. Sum = a / (1 - r).
About this calculator
Quick Reference
Ratio Testlimit of |aₙ₊₁/aₙ| < 1
Root Testlimit of ⁿ√|aₙ| < 1
Integral TestCompare to ∫f(x)dx
AbsoluteΣ|aₙ| also converges
Approaching a Limit
Infinite series are the sum of the terms of an infinite sequence. A series converges if the sum of its terms approaches a specific, finite value as more terms are added. If the sum grows without bound or fluctuates forever, it diverges.
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Pro Tips
The Geometric Series test is the most common: Σ arⁿ⁻¹ converges if |r| < 1.
The p-Series test: Σ 1/nᵖ converges if p > 1 (like 1/n²).
The Divergence Test: If the limit of individual terms is not zero, the series MUST diverge.
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Fun Facts
"The harmonic series (1 + 1/2 + 1/3 + ...) diverges, even though the terms get smaller and smaller!"
"Zeno's paradox (walking half the distance forever) is actually a convergent geometric series."
"The sum of all natural numbers (1 + 2 + 3 + ...) is formally divergent, but in certain physics contexts (string theory), it's associated with -1/12."
