What a Geometric Series Is
A geometric series is an infinite or finite sum in which each term is obtained by multiplying the previous term by the same constant number. That constant is called the common ratio.
For example, consider . Each term is obtained by multiplying the previous term by , so the series is geometric.
The most important skill is recognizing the common ratio. If the ratio between consecutive terms is constant, you have a geometric series.
Finding the Common Ratio
For a geometric sequence or series, divide any term by the term immediately before it:
For the series , for example, the common ratio is:
A negative common ratio is completely valid. It simply means the terms alternate signs, as in .
The Geometric-Series Formula
If a geometric series starts with first term and has common ratio , its finite partial sum through terms is:
This formula is useful for finite geometric sums. But AP Calculus BC frequently asks about infinite geometric series, where convergence becomes the key question.
When Does an Infinite Geometric Series Converge?
An infinite geometric series has the form . The crucial condition is the absolute value of the common ratio.
If , the powers of approach zero, so the partial sums approach a finite number. If , the terms do not approach zero, and the series cannot converge.
The Infinite Geometric-Series Sum
When , the infinite geometric series has the sum:
This formula is one of the most important formulas in the AP Calculus BC series unit. The two things you must check are:
- Identify the first term .
- Identify the common ratio and verify that .
Worked Examples
Find the sum of the infinite geometric series .
Determine whether converges. If it does, find its sum.
Determine whether the series converges.
Find the sum of .
Partial Sums and the Limit
The infinite-series formula is not a completely separate idea from the finite-sum formula. An infinite series is understood through the limit of its partial sums.
For a geometric series, the partial sum is . If , then approaches zero as approaches infinity. Therefore:
This is the mathematical reason the infinite geometric-series formula works.
Geometric Series in AP Calculus
Geometric series appear throughout the AP Calculus BC curriculum. They are especially important when you need to determine whether a series converges or find the exact value of an infinite sum.
A common AP-style pattern is a series written with a power of a variable:
For a particular value of , you can often identify the first term and common ratio immediately. This can turn what looks like a complicated infinite series into a one-line geometric-series calculation.
Common Mistakes
The formula only works when . Always check the common ratio before calculating a sum.
The common ratio is obtained by dividing a term by the previous term. For , the ratio is , not .
A negative ratio does not automatically mean divergence. For example, converges because . The signs alternate, but the terms shrink toward zero.
A sequence is a list of terms, while a series is a sum of terms. The geometric sequence generates the geometric series .
Practice Problems
Find the sum of .
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Determine whether converges or diverges. If it converges, find its sum.
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Find the sum of .
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For what values of does the geometric series converge?
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