Working with Geometric Series

Use the actual first included term and ratio, verifying convergence before applying a geometric sum.

College Board Topic 10.2 · CED p. 182 (PDF p. 187) · CED effective Fall 2020; project course outline 2026–27.

A geometric series has a constant ratio between consecutive terms. With first term a and ratio r, its terms are $a, a r, a r^{2}, \dots$ . The finite N-term sum is $S_{N} = a (1 - r^{N}) / (1 - r)$ when r≠1. Notice that N terms end at exponent N−1 if the first exponent is zero.

When $| r | < 1$ , rᴺ tends to zero, so ${\sum_{n = 0}}^{\infty} a r^{n} = a / (1 - r)$ . For a nonzero first term, ratios with |r|≥1 do not give a convergent infinite series: the terms do not tend to zero. If a=0 the series is trivially zero, regardless of r. A negative ratio alternates signs and still converges when its magnitude is less than 1.

The first term is the first included term, not always the coefficient printed before the power. In ${\sum_{n = 3}}^{\infty} 5 (1 / 5)^{n}$ , the first term is 1/25, and the sum is $(1 / 25) / (1 - 1 / 5) = 1 / 20$ . Check the starting index before using the formula.

Geometric series can model repeated shrinking contributions. Describe what each term counts: in a bouncing-ball distance, an initial drop is counted once and each subsequent bounce height is counted once up and once down. Read the retained geometric-series lesson for more practice.

Let's work through it

For $3 - 3 / 2 + 3 / 4 - \dots$ , identify $a = 3$ , $r = - 1 / 2$ . Why: Each successive term is −1/2 times the previous one.
Verify $| r | = 1 / 2 < 1$ . Why: The infinite geometric-sum formula requires this inequality.
The sum is $3 / (1 + 1 / 2) = 2$ . Why: Take the partial-sum limit, or use a/(1−r) after checking convergence.

Try it from memory

Pause & retrieve

Try it before you peek.

Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.

Question 1 of 20 self-assessed

Question 1

Does

{\sum_{n = 0}}^{\infty} 2 (- 1)^{n}

converge?

Your answer and reasoning

JavaScript enables the written-work controls. You can still try the questions below on paper and then open their reasoning.

Question 1: prompt and worked reasoning

Does

{\sum_{n = 0}}^{\infty} 2 (- 1)^{n}

converge?

No.

The terms do not tend to zero and the partial sums alternate.

Question 2: prompt and worked reasoning

What is the first term of

{\sum_{n = 4}}^{\infty} 3 (1 / 2)^{n}

3 / 16

Substitute the actual starting index n=4.

See every recall question with its answer

These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.

Retrieval 1

Does

{\sum_{n = 0}}^{\infty} 2 (- 1)^{n}

converge?

Worked answer and reason

No.

The terms do not tend to zero and the partial sums alternate.

Retrieval 2

What is the first term of

{\sum_{n = 4}}^{\infty} 3 (1 / 2)^{n}

Worked answer and reason

3 / 16

Substitute the actual starting index n=4.

Catch the tempting mistake

The coefficient a is always the first included term.: An offset starting exponent changes the first term. Evaluate it before finding the sum.

Calculator workspace and limits

For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.

Scientific calculator

This calculator requires JavaScript. Use the physics formula sheets and show your working manually.

Calculation is unavailable without JavaScript.

+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.

Put it to work

Unit 10 problem 3 · No calculator · analytic
Unit 10 problem 4 · No calculator · verbal

For the teacher

Use a finite geometric sum first. Then ask what must happen to rᴺ for the infinite limit to exist.

Unit planning and formative guidance →

Glossary and deeper reading · 3 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Convergence of a series
A series converges when the sequence of its partial sums approaches a finite limit. It diverges when there is no such finite limit. For example, 1/2+1/4+1/8+⋯ converges to 1, while the harmonic series diverges. Convergence of the individual terms to 0 is necessary but does not by itself establish convergence of their sum.
Read more: 10.1 · Convergence of a series →
Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.
Read more: 10.1 · Infinite series →
Index of a sequence or sum
An index identifies the position of a term in a sequence or finite or infinite sum. The starting index determines which term is included first. For example, a sum indexed by n=0 starts with the term obtained by substituting 0 into its term formula. Reindexing is valid only when the formula and bounds change together so that the same terms are retained.

Let's work through it

Try it from memory

Try it before you peek.

Question 1

Retrieval 1

Retrieval 2

Catch the tempting mistake

Put it to work

Convergence of a series

Infinite series

Index of a sequence or sum

Help us improve Omniguides