Sum of Infinite Geometric Series Calculator
calculator, Sum of Infinite Geometric series where r < 1

Calculates Sum of Infinite Geometric Series. When the first and ratio, r < 1 terms are given.$S_{\propto}=\frac{a}{1 - r}$

# sum of infinite geometric seriescalculator

The sum of infinite geometric series calculator determines the total sum of an infinite series that follows a geometric pattern, where the ratios between consecutive terms have an absolute value less than one.

### first term

Enter the first term of the GP series

### ratio

enter the ratio, ensure it is less than 1

### sum of infinite geometric series result

Hit the check mark to get the result

### About MathCrave Sum of Infinite Geometric Series Calculator?

Let's say we have a geometric series with a first term, a, and a common ratio, r. The formula for finding the sum of this infinite geometric series is:

• S = a / (1 - r),

where S represents the sum of the series.

To understand why this formula works, let's manipulate it a bit. Multiply both sides of the equation by (1-r):

• S(1 - r) = a.

Expanding the left side, we get:

• S - rS = a.

Rearranging the terms:

• S = a / (1 - r).

This formula allows us to find the sum of an infinite geometric series.

#### What Happens if Ratio is Less Than 1

However, there is a condition that must be met for this formula to work - the absolute value of the common ratio, |r|, must be less than 1.

If the common ratio has an absolute value less than 1, it means that each subsequent term in the series gets progressively smaller, tending towards zero. As a result, the sum of the series will converge to a finite value.

#### Simple Illustration of Sum of Infinite Geometric Series Where r <1

To understand this, imagine a geometric series where the common ratio is, for example, 0.5.

• The first term is a, the second term is a 0.5, the third term is a 0.5^2, and so on.

As the exponent increases, the value of 0.5^2 becomes smaller and smaller (0.5^2 < 0.5^1 < 0.5^0). If we continue this pattern indefinitely, the terms will approach zero.

The sum of an infinite geometric series with a common ratio less than 1 can be thought of as the limit of the series as the number of terms approaches infinity. This limit will yield a finite value.

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