Sequences and Series

How to find the sum of an infinite geometric series?

Q: What happens if the common ratio r is exactly 1 or -1?

If r = 1, each term is identical, so the series adds the same value infinitely and grows without bound. If r = -1, the partial sums oscillate between two numbers and never settle on a single limit. In both cases, the series diverges.

Q: Can an infinite geometric series have a negative sum?

Yes. If the first term a is negative and |r| < 1, the denominator (1 - r) remains positive, resulting in a negative total sum.

Q: Why does the formula S = a / (1 - r) work?

The sum of the first n terms is Sn = a(1 - r^n) / (1 - r). When |r| < 1, r^n approaches 0 as n goes to infinity, leaving only a / (1 - r).

Q: How is this topic practiced with Lessi?

Lessi is an AI teacher that guides you step by step through geometric series. The first 60 minutes of lessons with Lessi are free with an account.

Quick answer

You find the sum of an infinite geometric series using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This calculation only works when the common ratio satisfies |r| < 1, which allows the series to converge. If |r| ≥ 1, the series diverges and has no finite sum.

The topic explained

When does an infinite geometric series have a sum?

A geometric series multiplies each term by a fixed ratio, r, to get the next term. When you add infinitely many terms together, you might expect the sum to grow toward infinity. However, if each step adds a smaller and smaller amount, the total approaches a fixed number.

Before attempting to evaluate an infinite geometric series, always check the common ratio r first. The series converges only when the absolute value of r is strictly less than 1 (-1 < r < 1). If r is 1, -1, or outside this interval, the series diverges and has no finite sum.

Using the sum formula step by step

Once you confirm that |r| < 1, calculate the sum with the formula S = a / (1 - r), where a is the initial term.

Step 1: Identify the first term, a. Step 2: Divide any term by the preceding term to find r. Step 3: Check that -1 < r < 1. Step 4: Substitute a and r into S = a / (1 - r) and simplify.

For example, consider 8 + 4 + 2 + 1 + ... The first term is a = 8. Dividing 4 by 8 gives r = 0.5. Because |0.5| < 1, the series converges. Substitute these numbers into the formula: S = 8 / (1 - 0.5) = 8 / 0.5 = 16.

Converting repeating decimals to fractions

A repeating decimal is an infinite geometric series in disguise. Take 0.444... You can write this as 0.4 + 0.04 + 0.004 + ...

Here, the first term is a = 0.4 and the common ratio is r = 0.1. Because |0.1| < 1, use the sum formula: S = 0.4 / (1 - 0.1) = 0.4 / 0.9 = 4/9.

Free printable worksheet

Infinite Geometric Series and Convergence Practice

For 11th grade students. Determine whether each series converges, and find the sum or requested variable.

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Free printable worksheetlessoons.com

Infinite Geometric Series and Convergence PracticeName:

Determine whether the series 12 + 6 + 3 + ... converges. If so, find its sum.
Determine whether the series 5 - 10 + 20 - 40 + ... converges.
Find the sum of the series 27 - 9 + 3 - 1 + ...
Write the repeating decimal 0.777... as an infinite geometric series and calculate its fraction form.
An infinite geometric series has a sum of 40 and a common ratio of 3/4. Find its first term a.
The first term of an infinite geometric series is 15 and its sum is 20. Find the common ratio r.

Common questions

What happens if the common ratio r is exactly 1 or -1?

If r = 1, each term is identical, so the series adds the same value infinitely and grows without bound. If r = -1, the partial sums oscillate between two numbers and never settle on a single limit. In both cases, the series diverges.

Can an infinite geometric series have a negative sum?

Yes. If the first term a is negative and |r| < 1, the denominator (1 - r) remains positive, resulting in a negative total sum.

Why does the formula S = a / (1 - r) work?

The sum of the first n terms is Sn = a(1 - r^n) / (1 - r). When |r| < 1, r^n approaches 0 as n goes to infinity, leaving only a / (1 - r).

How is this topic practiced with Lessi?

Lessi is an AI teacher that guides you step by step through geometric series. The first 60 minutes of lessons with Lessi are free with an account.

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