Sequences & Series Calculator — AP, GP, Fibonacci & Summation
Sequences & Series Calculator
Explore arithmetic and geometric progressions, generate Fibonacci numbers, compute series sums,
and test convergence. All calculations are performed locally in your browser.
Mode:
Arithmetic Progression (AP)
Enter a1, d, n and click an operation
Formula: an = a1 + (n-1)d | Sn = n/2[2a1 + (n-1)d] = n/2[a1 + an]
Geometric Progression (GP)
Enter a1, r, n and click an operation
Formula: an = a1 · rn-1 | Sn = a1(1-rn)/(1-r) | S∞ = a1/(1-r) for |r| < 1
Tests: Geometric converges if |r| < 1. p-series converges if p > 1. Harmonic diverges.
Ratio test: if lim |an+1/an| < 1 then converges.
Sequences vs series
A sequence is an ordered list of terms; a series is what you get by adding them up. The two workhorse families: arithmetic progressions step by a constant +d (2, 5, 8, 11…) and geometric progressions multiply by a constant r (3, 6, 12, 24…).
An infinite geometric series can be summed exactly — a + ar + ar² + … = a/(1−r) — but only when |r| < 1; at |r| ≥ 1 the terms refuse to shrink and no finite total exists, which is what every convergence test is really checking.
