Advanced Geometric Series Calculator 2025

Calculate sums, specific terms, and visualize the series progression.

Series Parameters

Sum of First n Terms (Sₙ)

0

nth Term (aₙ)

0

Sum to Infinity (S∞)

0

Partial Sums of the Series

First 10 Terms of the Sequence

Geometric Series FAQ

What is a Geometric Series?

A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

What are the key formulas?

Let a be the first term, r be the common ratio, and n be the number of terms.

  • nth Term (aₙ): a * rⁿ⁻¹
  • Sum of first n terms (Sₙ): a * (1 - rⁿ) / (1 - r)
  • Sum to Infinity (S∞): a / (1 - r), only if |r| < 1.
What does it mean for a series to converge?

A geometric series "converges" if its sum approaches a specific, finite value as the number of terms goes to infinity. This only happens when the absolute value of the common ratio |r| is less than 1. If |r| is 1 or greater, the sum will either grow infinitely large or oscillate, and it is said to "diverge."