Geometric Series

Definition of Geometric Series

The series $\sum_{n=0}^{\infty} ar^n$ is called a geometric series, where $a \neq 0$ .

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\sum$	Greek letter	Sigma	Summation symbol, representing series
$\infty$	Mathematical symbol	Infinity	Represents infinite series, infinite number of terms
$r$	Mathematical symbol	Common ratio	Ratio between consecutive terms in geometric series

Convergence

Geometric Series Convergence

When $|r| < 1$ , the series converges, and its sum is:

$\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}$

When $|r| \geq 1$ , the series diverges.

证明

Let $S_n = a + ar + ar^2 + \cdots + ar^{n-1}$ , then multiply by $r$ to get: $rS_n = ar + ar^2 + \cdots + ar^n$
Subtract the two equations: $(1 - r)S_n = a - ar^n = a(1 - r^n)$
If $|r| < 1$ , then $\lim_{n \to \infty} r^n = 0$ , so: $\lim_{n \to \infty} S_n = \frac{a}{1 - r}$
When $|r| \geq 1$ , the partial sums do not converge, hence the series diverges.

Examples

Example 1

Determine the convergence of the series $\sum_{n=0}^{\infty} \frac{1}{2^n}$ and find its sum if it converges.

Solution: This is a geometric series with $a = 1, r = \frac{1}{2}$

Since $|r| = \frac{1}{2} < 1$ , the series converges.

The sum is: $S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2$

Example 2

Determine the convergence of the series $\sum_{n=0}^{\infty} \frac{1}{3^n}$ and find its sum if it converges.

Solution: This is a geometric series with $a = 1, r = \frac{1}{3}$

Since $|r| = \frac{1}{3} < 1$ , the series converges.

The sum is: $S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{3}} = \frac{3}{2}$

Example 3

Determine the convergence of the series $\sum_{n=0}^{\infty} 2^n$ .

Solution: This is a geometric series with $a = 1, r = 2$

Since $|r| = 2 \geq 1$ , the series diverges.

Exercises

Exercise 1

Determine the convergence of the series $\sum_{n=0}^{\infty} \frac{1}{4^n}$ and find its sum if it converges.

Reference Answer (3 个标签)

geometric series series convergence series summation

Problem-solving approach: This is a geometric series; we need to determine the relationship between the absolute value of the common ratio and 1.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{1}{4^n}$ is a geometric series
Determine parameters: $a = 1, r = \frac{1}{4}$
Check convergence: $|r| = \frac{1}{4} < 1$ , so the series converges
Calculate the sum: $S = \frac{a}{1-r} = \frac{1}{1-\frac{1}{4}} = \frac{4}{3}$

Answer: The series converges with sum $\frac{4}{3}$ .

Exercise 2

Determine the convergence of the series $\sum_{n=0}^{\infty} \frac{2}{5^n}$ and find its sum if it converges.

Reference Answer (3 个标签)

geometric series series convergence series summation

Problem-solving approach: This is a geometric series; we need to determine the relationship between the absolute value of the common ratio and 1.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} \frac{2}{5^n}$ is a geometric series
Determine parameters: $a = 2, r = \frac{1}{5}$
Check convergence: $|r| = \frac{1}{5} < 1$ , so the series converges
Calculate the sum: $S = \frac{a}{1-r} = \frac{2}{1-\frac{1}{5}} = \frac{2}{\frac{4}{5}} = \frac{5}{2}$

Answer: The series converges with sum $\frac{5}{2}$ .

Exercise 3

Determine the convergence of the series $\sum_{n=0}^{\infty} (-1)^n$ .

Reference Answer (2 个标签)

geometric series series convergence

Problem-solving approach: This is a geometric series; we need to determine the relationship between the absolute value of the common ratio and 1.

Detailed steps:

Identify the series type: $\sum_{n=0}^{\infty} (-1)^n$ is a geometric series
Determine parameters: $a = 1, r = -1$
Check convergence: $|r| = |-1| = 1 \geq 1$ , so the series diverges

Answer: The series diverges.

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$a$	Mathematical symbol	First term	First term of geometric series
$S_n$	Mathematical symbol	Partial sum	Sum of first $n$ terms of the series
$\lim$	Mathematical symbol	Limit	Represents limit of sequence or function

Chinese-English Glossary

Chinese Term	English Term	IPA Pronunciation	Explanation
几何级数	geometric series	/dʒiːəˈmetrɪk ˈsɪəriːz/	Series of the form $\sum_{n=0}^{\infty} ar^n$
公比	common ratio	/ˈkɒmən ˈreɪʃiəʊ/	Ratio between consecutive terms $r$ in geometric series
首项	first term	/fɜːst tɜːm/	First term $a$ of geometric series
收敛	convergence	/kənˈvɜːdʒəns/	Partial sums sequence has a finite limit
发散	divergence	/daɪˈvɜːdʒəns/	Partial sums sequence has no finite limit
和	sum	/sʌm/	Limit value of convergent series
部分和	partial sum	/ˈpɑːʃəl sʌm/	Sum of first $n$ terms $S_n$ of the series

PreviousConvergence TestsLearn various convergence tests for series, including tests for positive-term series, alternating series, and general series. Nextp-SeriesLearn the definition, convergence, and applications of p-series.

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