Difficulty: Intermediate | Prerequisites: Familiarity with sequences, partial sums, and the definitions of convergent and divergent series.
Geometric series sit at the crossroads of sequences and series in integral calculus. This topic builds on your understanding of how individual terms in a sequence behave and asks a bigger question: what happens when you add up infinitely many of them? If you are comfortable writing out the first several terms of a sequence and recognise how partial sums work, you are ready for this. The payoff is a clean, closed-form formula that lets you collapse an infinite sum into a single fraction, provided the series converges.
A geometric series adds up terms that each multiply the previous one by a fixed ratio r. If the absolute value of r is less than 1, the series converges and its infinite sum equals a/(1 - r), where a is the first term. If |r| is 1 or larger, the series diverges and has no finite sum.
Common ratio (r)
The fixed multiplier between consecutive terms of a geometric series. You find it by dividing any term by the one before it. In simple terms, it is the number you keep multiplying by to get the next term.
Infinite series
A sum that keeps going without end, written as the sum from n = 1 to infinity of a_n. Think of it as adding up every single term in a sequence, one after another, forever.
Partial sum (S_n)
The sum of the first n terms of a series. In simple terms, it is what you get when you stop adding after a finite number of terms and check the running total.
Convergence
A series converges when its partial sums approach a specific, finite number as n grows without bound. Think of it as the sum "settling down" to a fixed value.
Divergence
A series diverges when its partial sums do not approach any finite limit. The total either blows up, oscillates, or otherwise refuses to settle.
Geometric series
A series in which each term is obtained by multiplying the previous term by a constant ratio r. The general form is a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio.
An infinite series is the sum of all terms in an infinite sequence: a_1 + a_2 + a_3 + ... + a_n + ...
The first step with any series is to spot the pattern. Look at how each term relates to the one before it.
1/2 + 1/4 + 1/8 + ... Each term is half the previous one. Common ratio r = 1/2. In sigma notation: sum from n = 1 to infinity of (1/2)^n.
2 + 3/2 + 9/8 + 27/32 + ... The first term is 2 and each subsequent term multiplies by 3/4. In sigma notation: sum from n = 1 to infinity of 2(3/4)^(n - 1).
Not every infinite series is geometric. The series 1/2 + 2/3 + 3/4 + ... has nth term n/(n + 1), which is not a constant ratio between terms. This one is not geometric and requires different tools.
A partial sum S_n is the result of adding the first n terms and stopping.
You can compute partial sums on a calculator using the summation template. For example:
Sum from n = 1 to 8 of (1/3)^n = 3280/6561, which is roughly 0.4999.
Sum from n = 1 to 6 of (1/2)^n = 63/64 = 0.984375.
Sum from n = 1 to 6 of 2(3/4)^(n - 1) = 3367/512, which is roughly 6.576.
As you add more terms (increase n), watch what happens to S_n. If it levels off, the series is converging. If it keeps growing or oscillating, it is diverging.
A geometric series sum from n = 1 to infinity of ar^(n - 1) converges if and only if |r| < 1.
When it converges, the sum equals a/(1 - r).
When |r| >= 1, the series diverges. There is no finite sum.
A subtle point: when r is negative and |r| >= 1 (for example, r = -3/2), the series does not blow up to infinity. Instead, the partial sums oscillate between positive and negative values, growing in magnitude. It still diverges, just not "to infinity" in the usual sense.
Identify the first term a and the common ratio r.
Confirm |r| < 1.
Plug into S = a/(1 - r).
Example: sum from n = 1 to infinity of (1/2)^n. Here a = 1/2 and r = 1/2. Sum = (1/2)/(1 - 1/2) = 1.
Example: sum from n = 1 to infinity of 2(3/4)^(n - 1). Here a = 2 and r = 3/4. Sum = 2/(1 - 3/4) = 2/(1/4) = 8.
Example: sum from n = 1 to infinity of (2/3)(3/2)^(n - 1). Here a = 2/3 and r = 3/2. Since |r| = 3/2 >= 1, the series diverges. No sum exists.
One practical approach: generate the first 50 partial sums in a list and plot them as a scatter plot.
For sum of (1/2)^n, the scatter plot of partial sums levels off at 1, visually confirming convergence.
For sum of 2(3/4)^(n - 1), partial sums level off near 8.
For sum of (2/3)(3/2)^(n - 1), partial sums grow without bound, confirming divergence.
This graphical method is useful for building intuition before applying the formula.
For the series sum of (1/2)^n, list partial sums: S_1 = 1/2, S_2 = 3/4, S_3 = 7/8, S_4 = 15/16, ...
Notice the pattern: the denominator is 2^n and the numerator is always one less, giving S_n = (2^n - 1)/2^n.
Take the limit as n approaches infinity: lim (2^n - 1)/2^n = 1.
This confirms the convergence and gives the sum directly from the pattern in the partial sums.
For a geometric series, the ratio of consecutive terms a_(n+1)/a_n equals r for every n.
If |r| < 1, terms shrink toward zero and the series converges.
If |r| >= 1, terms do not shrink (they grow or stay the same size), and the series diverges.
This is the basis for the geometric series convergence test: convergence requires |r| < 1.
The formula works the same way regardless of starting index, but you must identify a correctly.
Starting at n = 0: sum from n = 0 to infinity of ar^n = a/(1 - r).
Starting at n = 1: sum from n = 1 to infinity of ar^(n - 1) = a/(1 - r).
Both forms give the same result. The key is that a is always the first term of the series.
\sum_{n=1}^{\infty} ar^{n-1} = \frac{a}{1-r}, \quad |r| < 1S_n = a \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1\text{Convergence condition: } |r| < 1The first formula gives the infinite sum. The second gives the partial sum of the first n terms and is useful for computing specific values or spotting the pattern that leads to the infinite sum. The convergence condition is the gate: check it before applying the infinite sum formula.
Geometric series appear whenever a quantity decays by a fixed fraction each period. A bouncing ball that reaches 3/4 of its previous height on every bounce travels a total distance described by a geometric series. In finance, the present value of a perpetuity (an infinite stream of fixed payments discounted at a constant rate) is calculated using the same a/(1 - r) formula. Signal processing, population modelling with constant decay rates, and drug dosage accumulation in pharmacology all rely on convergent geometric series.
Students often think that if individual terms approach zero, the series must converge. It does not. Terms approaching zero is necessary for convergence but not sufficient. The harmonic series (1 + 1/2 + 1/3 + ...) has terms going to zero, yet it diverges.
Students sometimes confuse the common ratio r with the first term a. The ratio is the multiplier between consecutive terms, not the starting value.
When r is negative, students may assume the series "converges" because terms alternate in sign. A negative r with |r| >= 1 still produces a divergent series, even though it does not blow up to positive infinity.
Students frequently misidentify a when the summation starts at n = 0 versus n = 1. The first term a is whatever the series actually produces when you plug in the starting index, not always the coefficient sitting in front of r.
⚠️ You will almost certainly be asked to determine whether a given geometric series converges or diverges. The answer is entirely determined by |r|: less than 1 means convergence, 1 or greater means divergence.
⚠️ Expect to compute the exact sum using a/(1 - r). Make sure you correctly identify a (the first term) and r (the common ratio) before substituting.
⚠️ Questions may present the series starting at n = 0 or n = 1. Both lead to the same formula, but misreading the index will give you the wrong first term.
⚠️ You may be asked to find a partial sum S_n and then take its limit. Know the partial sum formula S_n = a(1 - r^n)/(1 - r) and be comfortable taking lim as n approaches infinity.
⚠️ Graphical interpretation: if asked to sketch or describe the behaviour of partial sums, a convergent geometric series produces a scatter plot that levels off at the sum. A divergent one grows without bound (or oscillates for negative r with |r| >= 1).
True or false: A geometric series with r = -0.5 converges. (True. |r| = 0.5 < 1.)
Fill in the blank: The infinite sum of 3(1/4)^(n - 1) from n = 1 to infinity equals ___. (4. Using a/(1 - r) = 3/(1 - 1/4) = 3/(3/4) = 4.)
True or false: If every term of a series approaches zero, the series must converge. (False. The harmonic series is a counterexample.)
Fill in the blank: A geometric series converges if and only if |r| is ___ than 1. (Less.)
True or false: The series sum of 5(2)^(n - 1) from n = 1 to infinity converges to 10. (False. r = 2, so |r| >= 1 and the series diverges.)
Q: Determine whether the series sum from n = 1 to infinity of (1/3)^n converges or diverges. If it converges, find the sum.
A: The common ratio is r = 1/3. Since |1/3| < 1, the series converges. The first term (when n = 1) is 1/3. Sum = (1/3)/(1 - 1/3) = (1/3)/(2/3) = 1/2.
Q: Find the sum of the geometric series sum from n = 1 to infinity of 2(3/4)^(n - 1).
A: Here a = 2 and r = 3/4. Since |3/4| < 1, the series converges. Sum = 2/(1 - 3/4) = 2/(1/4) = 8.
Q: Does the series sum from n = 1 to infinity of (2/3)(3/2)^(n - 1) converge or diverge? Explain.
A: The common ratio is r = 3/2. Since |3/2| = 1.5 >= 1, the series diverges. No finite sum exists.
Q: Write the first six partial sums of the series sum from n = 1 to infinity of (1/2)^n and identify the pattern.
A: S_1 = 1/2, S_2 = 3/4, S_3 = 7/8, S_4 = 15/16, S_5 = 31/32, S_6 = 63/64. The pattern is S_n = (2^n - 1)/2^n. As n approaches infinity, this approaches 1.
Q: A geometric series has a first term of 5 and converges to 20. What is the common ratio?
A: Using S = a/(1 - r), we have 20 = 5/(1 - r). Solving: 1 - r = 5/20 = 1/4, so r = 3/4.
Q: Explain why the condition |r| < 1 is necessary for convergence of a geometric series.
A: When |r| < 1, each successive term r^n shrinks toward zero, so the added contributions become negligible and the partial sums level off at a finite value. When |r| >= 1, the terms do not shrink (they stay the same size or grow), so the partial sums cannot settle to a finite limit.
Geometric series are foundational for power series and Taylor series, which generalise the idea of representing functions as infinite sums. If you can identify and sum geometric series confidently, Taylor series will feel like a natural extension. This topic also connects to the ratio test for general series: the ratio test is essentially asking whether a series "behaves like" a convergent geometric series for large n. In applications courses, the present value of annuities and perpetuities in finance, and exponential decay models in physics and biology, both rest directly on this material.
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