Infinite Sum Calculator

Solve infinite geometric series instantly. Determine convergence, calculate sum limits, and visualize the progression toward a finite total.

Geometric Series Parameters

For convergence, |r| must be less than 1.

Result
Convergent
S = a / (1 - r)
2
Partial Sums Approximation
n=1 1.0000
n=5 1.9375
Convergence Visualization

What is an Infinite Sum?

An infinite sum is the total value of adding all terms in an infinite sequence. In mathematics, this is often represented using the Greek letter Sigma. While it might seem impossible to add an infinite number of things together and get a finite number, many series do exactly that through a process called convergence.

The most common type of infinite series that can be summed is the geometric series. In this series, each term is found by multiplying the previous term by a fixed number called the common ratio.

Key Terms

  • ● First Term: The starting number of the series.
  • ● Common Ratio: The constant factor between terms.
  • ● Convergent: A series that approaches a finite sum.
  • ● Divergent: A series that goes to infinity and has no finite sum.

Frequently Asked Questions

Common questions about infinite series and convergence.

The sum of an infinite geometric series is given by the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. This formula only applies when the absolute value of 'r' is less than 1.

If |r| ≥ 1, the series is divergent. This means the terms do not get smaller fast enough (or they get larger), and the total sum tends toward infinity rather than a finite number.

Yes. If the first term 'a' is negative and the series converges, the infinite sum will be negative. Similarly, if 'r' is negative, the terms will alternate, but the sum will still be finite if |r| < 1.

A series is convergent if the sequence of its partial sums approaches a specific, finite limit as the number of terms goes to infinity.