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We explore the **geometric series test**, a cornerstone concept in **mathematical sequences** and **series**. This article will delve into the **theory**, **proofs**, and **applications** of this influential test.

The **geometric series test** offers a gateway to understanding whether an **infinite geometric series** **converges** or **diverges**, providing a solid foundation for subsequent **mathematical theories**.

Whether you’re a seasoned **mathematician**, a budding **student**, or a curious **reader**, this exploration will illuminate new facets of **mathematics**, emphasizing its **elegance**, **rigor**, and **practical relevance**. Join us as we navigate the nuances of this fascinating topic, shedding light on its intriguing implications and **potential applications**.

## Definition of Geometric Series Test

The **geometric series test** is a **mathematical method** to determine whether a given **geometric series** **converges** or **diverges**. A geometric series is a **sequence** of terms in which each **subsequent term** after the first is found by multiplying the previous term by a fixed, **non-zero number** called the **common ratio**.

The test states that a **geometric series** ∑$r^n$ (where n runs from 0, 1, 2, up to ∞) will **converge** if the **absolute value** of r is less than 1 (**|r| < 1**) and will **diverge** otherwise. When it converges, the **sum** of the geometric series can be found using the formula **S = a / (1 – r)**, where **‘a’** is the **first term** and **‘r’** is the **common ratio**.

Below we present a generic representation of the geometric series in continuous and discrete form in figure-1.

Figure-1.

## Historical Significance

The concept of **geometric series** has been known since **ancient times**, with early evidence of its use found in both **Greek** and **Indian mathematics**.

The **ancient Greeks** were among the first to explore **geometric series**. The philosopher **Zeno of Elea**, famous for his paradoxes, devised a series of thought experiments that implicitly relied on geometric series, particularly his “**dichotomy paradox**,” which essentially describes a geometric series where the common ratio is 1/2.

Indian **mathematicians**, notably in the classical age around **5th** to **12th century AD**, made substantial contributions to the understanding of **geometric progressions** and **series**. A key figure in this development was **Aryabhata**, an Indian mathematician and **astronomer** from the late **5th** and early **6th century**, which used **geometric series** to give a formula for the sum of finite geometric series and applied it to compute interest.

The understanding of the **geometric series** evolved significantly in the late **Middle Ages**, particularly with the work of **medieval Islamic mathematicians**. They used **geometric series** to solve **algebraic problems** and offered explicit formulas for the sum of **finite geometric series**.

However, it wasn’t until the **17th century** and the advent of **calculus** that mathematicians studied the **convergence** and **divergence** of infinite series more systematically. The understanding of **geometric series**, including the **convergence criterion** (**|r| < 1** for convergence), was deepened with the work of mathematicians like **Isaac Newton** and **Gottfried Wilhelm Leibniz**, the co-founders of **calculus**.

The **geometric series test**, as it is understood today, is essentially the culmination of centuries of accumulated knowledge, stretching back to the ancient **Greeks** and **Indians**, through the Islamic mathematicians of the **Middle Ages**, up to the mathematical pioneers of the Age of **Enlightenment**. Today, it remains a fundamental concept in mathematics, **underpinning** many areas of study and application.

**Properties**

**Convergence Criterion**

The **geometric series test** states that the geometric series, **∑a*$r^n$** **converges** if and only if the absolute value of the **common ratio** is less than** 1 (|r| < 1)**. If **|r| >= 1**, the series does not converge (i.e., it **diverges**).

**Sum of Converging Geometric Series**

If the **geometric series converges**, its sum can be calculated using the formula **S = a / (1 – r)**, where **‘S’** represents the** sum** of the series, **‘a’** is the first term, and** ‘r’** is the **common ratio**.

**The behavior of the Series**

For **|r| < 1**, as n approaches **infinity**, the terms in the series approach** zero**, meaning the series **“settles”** to a finite number. If **|r| >= 1**, the terms in the series do not approach zero, and the series **diverges**, meaning it doesn’t settle for a **finite** value.

**Negative Common Ratio**

If the **common ratio ‘r’** is** negative** and its **absolute** value is less than** 1** (i.e., -1 < r < 0), the series still **converges**. However, the terms of the series will **oscillate** between positive and negative values.

**Independent of First Term**

The **convergence** or **divergence** of a **geometric series** does not depend on the value of the first term** ‘a’**. Regardless of the value of **‘a’**, if **|r| < 1**, the series will **converge**, and if **|r| >= 1**, it will **diverge**.

**Partial Sums:** The partial sums of a geometric series form a **geometric sequence t**hemselves. The **n-th** p**artial sum** of the series is given by the formula **$S_n$ = a * (1 – $r^n$) / (1 – r)** for **r ≠ 1**.

**Applications **

The **geometric series test** and the principles of geometric series find applications across a broad range of fields, from pure **mathematic**s to** physics**, **economics**, **computer science,** and even in **biological modeling**.

**Mathematics**

The concept of **geometric series** is **instrumental** in **calculus** and is frequently used in **conjunction** with** power series** or **Taylor series**. They can also be used to solve **difference equations**, which have applications in **dynamic systems**, like **population modeling**, where the change in population from year to year follows a **geometric pattern**.

**Physics**

In **electrical engineering**, the principles of **geometric series** can be used to calculate the equivalent resistance of an infinite number of resistors arranged in **parallel** or in **series**. In **optics**, geometric series can be used to analyze the behavior of light as it repeatedly reflects between two **parallel mirrors**.

**Computer Science**

Concepts from **geometric series** are often found in the design and **analysis o**f **algorithms**, especially those with recursive elements. For example, **binary search algorithms**, **divide and conquer algorithms**, and algorithms dealing with data structures like **binary trees** often involve geometric series in their **time complexity analysis**.

**Economics and Finance**

**Geometric series** find extensive use in calculating the present and future values of **annuities** (fixed sum paid every year). They’re also used in models of **economic growth** and the study of functions of **compounded interest**. Furthermore, they’re utilized to evaluate **perpetuities** (an infinite sequence of cash flows).

**Biology**

**Geometric series** can be used in biological modeling. In **population modeling**, for instance, the size of each generation might be modeled as a** geometric series**, assuming each generation is a fixed multiple of the size of the previous one.

**Engineering**

In **control theory**, g**eometric series** can be used to analyze systems’ responses to certain **inputs**. If a system’s output at any given time is a **proportion** of its input at the previous time, the total response over time forms a** geometric series**.

**Probability Theory and Statistics**

In a **geometric distribution**, the number of trials needed to get the first success in a series of **Bernoulli trials** is modeled. Here, the **expected value a**nd **variance** of a **geometric distribution** are derived using **geometric series**.

**Exercise **

**Example 1 **

Determine if the series **∑$(2/3)^n$** from **n=0** to **∞** **converges **or** diverges**.

**Solution**

In the series **∑$(2/3)^n$**, the common ratio** r = 2/3**. Since the absolute value of **r**, **|r| = |2/3| = 2/3**, which is less than **1**, the geometric series **converges** according to the **geometric series test**.

Figure-2.

**Example 2**

Determine the sum of the series **∑$(2/3)^n$** from **n=0** to **∞**.

**Solution**

Since the series **∑$(2/3)^n$** converges, we can find the sum of the series using the formula a / (1 – r), where **‘a’** is the first term and **‘r’** is the **common ratio**. Here, a = $(2/3)^0$ = 1, and r = 2/3. So, the sum is:

S = 1 / (1 – 2/3)

S = 1 / (1/3)

S = 3

**Example 3**

Determine if the series **∑$2^n$** from **n=0** to **∞** **converges **or** diverges**.

**Solution**

In the series **∑$2^n$**, the common ratio** r = 2**. Since the absolute value of **r**:

|r| = |2| = 2

which is greater than **1**, the geometric series diverges according to the **geometric series test**.

Figure-3.

**Example 4**

Determine the sum of the series **∑$(-1/2)^n$** from **n=0** to **∞**.

**Solution**

In the series **∑$(-1/2)^n$**, the common ratio** r = -1/2**. Since the absolute value of** r**,** |r| = |-1/2| = 1/2**, which is less than **1**, the geometric series converges according to the **geometric series test**.

Here:

a = $(-1/2)^0$

a = 1

and

r = -1/2

So, the sum is:

S = 1 / (1 – (-1/2))

S = 1 / (1.5)

S = 2/3

**Example 5**

Determine if the series **∑$(-2)^n$** from **n=0** to **∞** **converges** or **diverges**.

**Solution**

In the series **∑$(-2)^n$**, the common ratio** r = -2**. Since the absolute value of **r**, **|r| = |-2| = 2**, which is greater than **1**, the geometric series diverges according to the **geometric series test**.

**Example 6**

Determine the sum of the series **∑$0.5^n$** from **n=1** to **∞**.

**Solution**

In the series **∑$0.5^n$**, the common ratio **r = 0.5**. Since the absolute value of **r**, **|r| = |0.5| = 0.5**, which is less than **1**, the geometric series converges according to the **geometric series test**. Here:

a = $0.5^1$

a = 0.5

and

r = 0.5

So, the sum is:

S = 0.5 / (1 – 0.5)

S = 0.5 / 0.5

S = 1

**Example 7**

Determine if the series **∑$(5/4)^n$** from **n=1** to **∞** converges or diverges.

**Solution**

To determine if the series **∑$(5/4)^n$** from **n=1** to** ∞** converges or diverges, we need to examine the behavior of the **common ratio**.

The series can be written as:

∑$(5/4)^n$ = $(5/4)^1$ + $(5/4)^2$ + $(5/4)^3$ + …

The common ratio, denoted by r, is the ratio of consecutive terms. In this case, r = 5/4.

If the absolute value of the common ratio |r| is less than 1, the series converges. If |r| is greater than or equal to 1, the series diverges.

In this example, **|5/4| = 5/4 = 1.25**, which is greater than **1**. Therefore, the series diverges.

The series **∑$(5/4)^n$** from **n=1** to **∞** **diverges**.

**Example 8**

Determine the sum of the series **∑$(-1/3)^n$** from **n=0** to **∞**.

**Solution**

To determine the sum of the series** ∑$(-1/3)^n$** from n=0 to ∞, we can use the formula for the sum of a **convergent geometric series**.

The series can be written as:

∑$(-1/3)^n$ = $(-1/3)^0$ + $(-1/3)^1$ + $(-1/3)^2$ + …

The common ratio, denoted by **r**, is the ratio of consecutive terms. In this case,** r = -1/3**.

If the absolute value of the common ratio **|r|** is less than **1**, the series converges. If **|r|** is greater than or equal to **1**, the series **diverges**.

In this example, **|(-1/3)| = 1/3**, which is less than **1, **therefore, the series **converges**.

The sum of the series can be calculated using the formula:

a / (1 – r)

where a is the first term and r is the **common ratio**.

In this case:

a = $(-1/3)^0$

a = 1

and

r = -1/3

The sum is given by:

S = a / (1 – r)

S = 1 / (1 – (-1/3))

S = 1 / (1 + 1/3)

S = 1 / (4/3)

S = 3/4

S ≈ 0.75

Therefore, the sum of the series **∑$(-1/3)^n$** from **n=0** to **∞** is approximately **0.75**.

*All images were created with MATLAB.*