MTH120 College Algebra Chapter 9.4

 9.4 Series and Their Notations

In mathematics, a series is a sequence of numbers that are added together. The notation used to represent a series and its sum is essential for understanding and working with these mathematical concepts. Here are some key notations and concepts related to series:

1. Summation Notation (∑): The Greek letter sigma (∑) is used to represent summation. It indicates that you should add up a series of terms. The notation is followed by the index of summation (usually a variable like $�$) which ranges from a lower bound to an upper bound. For example, ${\sum }_{�=1}^{10}{�}_{�}$ means you should add up the terms ${�}_1$ through ${�}_{10}$.

2. General Term: The ${�}_{�}$ in the summation notation represents the general term of the series. It's a formula or rule that gives you the nth term in the series. For example, in the series ${\sum }_{�=1}^{10}2�$, the general term is $2�$, which gives the nth term.

3. Lower and Upper Bounds: The lower and upper bounds of the summation notation define the range of terms to be added. In ${\sum }_{�=1}^{10}2�$, the lower bound is 1, and the upper bound is 10, meaning you add up terms from ${�}_1$ to ${�}_{10}$.

4. Infinite Series: In some cases, a series may extend indefinitely. In such cases, the upper bound is represented as infinity (∞). An infinite series may or may not have a finite sum, depending on its convergence properties.

5. Partial Sum: The sum of a finite number of terms in a series is called the partial sum. The $�$th partial sum is denoted as ${�}_{�}$ and is found by adding the first $�$ terms of the series: ${�}_{�}={\sum }_{�=1}^{�}{�}_{�}$.

6. Convergence and Divergence: A series is said to converge if the sum of its terms approaches a finite value as more terms are added. If the sum does not approach a finite value, the series is said to diverge.

7. Geometric Series: A geometric series is a series in which each term is a constant multiple of the previous term. The sum of a finite geometric series is given by the formula $\frac{�(1-{�}^{�})}{1-�}$, where $�$ is the first term, $�$ is the common ratio, and $�$ is the number of terms.

8. Arithmetic Series: An arithmetic series is a series in which each term is obtained by adding a fixed constant to the previous term. The sum of a finite arithmetic series is given by the formula ${�}_{�}=\frac{�}{2}[2�+(�-1)�]$, where $�$ is the first term, $�$ is the number of terms, and $�$ is the common difference between terms.

These notations and concepts are fundamental to working with series in mathematics. They are used to express, analyze, and compute the sums of various types of series, making them essential tools in calculus, algebra, and many other areas of mathematics.

Summation notation, denoted by the Greek letter sigma (∑), is a concise way to represent the sum of a series of terms. It allows you to express the addition of a sequence of numbers with a compact notation. The notation is typically followed by the index variable, the lower bound, and the upper bound. Here are some examples of how to use summation notation:

1. Sum of Natural Numbers: The sum of the first 10 natural numbers (1, 2, 3, ..., 10) can be expressed using summation notation as follows: ${\sum }_{�=1}^{10}�$ This notation represents the sum of all values of $�$ from 1 to 10.

2. Sum of Squares: To represent the sum of the squares of the first 5 natural numbers, you can use the following summation notation: ${\sum }_{�=1}^5{�}^2$ This notation means you should add the squares of values from 1 to 5: $1^2+2^2+3^2+4^2+5^2$.

3. Sum of Even Numbers: If you want to represent the sum of the first 6 even numbers, you can use this summation notation: ${\sum }_{�=1}^{6}2�$ This means adding the first 6 even numbers: $2+4+6+8+10+12$.

4. Sum of a Sequence with a Formula: You can use summation notation to represent the sum of a sequence with a general term formula. For example, the sum of the first 7 terms of a sequence defined by ${�}_{�}=3�+2$ can be expressed as: ${\sum }_{�=1}^7(3�+2)$ This notation represents adding the first 7 terms of the sequence where each term is obtained by substituting $�$ into the formula $3�+2$.

5. Sum of an Infinite Series: Summation notation can also represent infinite series. For example, the sum of an infinite geometric series with the general term $\frac{1}{2^{�}}$ is expressed as: ${\sum }_{�=1}^{\mathrm{\infty }}\frac{1}{2^{�}}$ This notation indicates that you're adding an infinite number of terms in the series.

In each of these examples, summation notation simplifies the representation of series, making it more compact and convenient for expressing mathematical concepts and calculations. The lower and upper bounds of the summation indicate the range of terms to be added.

Summation notation, often represented by the Greek letter sigma (∑), is a mathematical shorthand used to express the sum of a sequence of terms. It's a compact way to write down the addition of a series of numbers or other expressions. The notation is typically followed by the index variable, the lower bound, the upper bound, and the expression to be summed. Here's a breakdown of the components:

1. Sigma (∑): This symbol indicates that a sum is to be taken. It is the starting point of the summation notation.

2. Index Variable: The index variable (often represented by a letter like $�$, $�$, or $�$) is a placeholder that iterates through the terms of the series. It starts at the lower bound and goes up to the upper bound.

3. Lower Bound: The lower bound specifies the starting point for the index variable. It defines where the summation begins.

4. Upper Bound: The upper bound sets the limit for the index variable. It defines where the summation ends.

5. Expression: The expression to the right of the sigma symbol represents the formula or rule for generating each term in the series. It typically depends on the index variable.

The general form of summation notation is:

${\sum }_{\text{index variable}=\text{lower bound}}^{\text{upper bound}}\text{expression}$

Here are a few examples to illustrate how summation notation works:

1. Sum of First 10 Natural Numbers: The sum of the first 10 natural numbers can be written as: ${\sum }_{�=1}^{10}�$ This notation means you add the values of $�$ from 1 to 10, resulting in $1+2+3+\dots +10$.

2. Sum of Squares of First 5 Natural Numbers: The sum of the squares of the first 5 natural numbers is expressed as: ${\sum }_{�=1}^5{�}^2$ This notation represents the sum $1^2+2^2+3^2+4^2+5^2$.

3. Sum of a Sequence with a Formula: You can use summation notation to represent the sum of a sequence defined by a formula. For example, the sum of the first 7 terms of a sequence defined by ${�}_{�}=3�+2$ is written as: ${\sum }_{�=1}^7(3�+2)$ This notation means you're adding the terms $5+8+11+14+17+20+23$.

Summation notation simplifies the representation of series, making it more concise and convenient for expressing mathematical concepts, calculations, and mathematical operations. It's commonly used in various branches of mathematics, including calculus, discrete mathematics, and statistics.

The formula for the sum of an arithmetic series is a handy tool for finding the sum of a sequence of numbers where each term is generated by adding a fixed constant (the common difference, $�$) to the previous term. The formula is based on the number of terms in the series, the first term (${�}_1$), and the common difference ($�$). Here's the formula:

Sum of an Arithmetic Series (${�}_{�}$):

${�}_{�}=\frac{�}{2}[2{�}_1+(�-1)�]$

Where:

• ${�}_{�}$ is the sum of the first $�$ terms of the series.
• $�$ is the number of terms.
• ${�}_1$ is the first term.
• $�$ is the common difference between terms.

Let's use this formula with some examples:

Example 1: Find the sum of the first 10 terms of the arithmetic series 5, 8, 11, 14, ...

• Number of terms ($�$): 10
• First term (${�}_1$): 5
• Common difference ($�$): 3 (each term is 3 greater than the previous one)

Using the formula:

${�}_{10}=\frac{10}{2}[2\cdot 5+(10-1)\cdot 3]=5\cdot [10+27]=5\cdot 37=185$

So, the sum of the first 10 terms of the series is 185.

Example 2: Find the sum of the first 15 terms of the arithmetic series -2, 1, 4, 7, ...

• Number of terms ($�$): 15
• First term (${�}_1$): -2
• Common difference ($�$): 3

Using the formula:

${�}_{15}=\frac{15}{2}[2\cdot (-2)+(15-1)\cdot 3]=\frac{15}{2}[-4+42]=\frac{15}{2}\cdot 38=285$

So, the sum of the first 15 terms of the series is 285.

Example 3: Find the sum of the first 20 terms of the arithmetic series 10, 7, 4, 1, ...

• Number of terms ($�$): 20
• First term (${�}_1$): 10
• Common difference ($�$): -3

Using the formula:

${�}_{20}=\frac{20}{2}[2\cdot 10+(20-1)\cdot (-3)]=10\cdot [20-57]=10\cdot (-37)=-370$

So, the sum of the first 20 terms of the series is -370.

The formula for the sum of an arithmetic series simplifies the process of finding the sum of a series with a known first term, common difference, and number of terms. It's a valuable tool for various mathematical and real-world applications.

The formula for the sum of the first $�$ terms of an arithmetic series, denoted as ${�}_{�}$, is as follows:

${�}_{�}=\frac{�}{2}[2{�}_1+(�-1)�]$

Where:

• ${�}_{�}$ is the sum of the first $�$ terms of the series.
• $�$ is the number of terms.
• ${�}_1$ is the first term of the series.
• $�$ is the common difference between terms.

Let's use this formula in a couple of examples:

Example 1: Find the sum of the first 8 terms of the arithmetic series 3, 7, 11, 15, ...

• Number of terms ($�$): 8
• First term (${�}_1$): 3
• Common difference ($�$): 4

Using the formula:

${�}_8=\frac{8}{2}[2\cdot 3+(8-1)\cdot 4]=4\cdot [6+7\cdot 4]=4\cdot (6+28)=4\cdot 34=136$

So, the sum of the first 8 terms of the series is 136.

Example 2: Find the sum of the first 12 terms of the arithmetic series -5, -2, 1, 4, ...

• Number of terms ($�$): 12
• First term (${�}_1$): -5
• Common difference ($�$): 3

Using the formula:

${�}_{12}=\frac{12}{2}[2\cdot (-5)+(12-1)\cdot 3]=6\cdot [-10+33]=6\cdot 23=138$

So, the sum of the first 12 terms of the series is 138.

Example 3: Find the sum of the first 20 terms of the arithmetic series 2, 5, 8, 11, ...

• Number of terms ($�$): 20
• First term (${�}_1$): 2
• Common difference ($�$): 3

Using the formula:

${�}_{20}=\frac{20}{2}[2\cdot 2+(20-1)\cdot 3]=10\cdot [4+57]=10\cdot 61=610$

So, the sum of the first 20 terms of the series is 610.

The formula for the sum of the first $�$ terms of an arithmetic series simplifies the process of finding the sum when you know the first term, common difference, and the number of terms. It's a valuable tool for various mathematical and real-world applications, including finance, physics, and engineering.

The formula for the sum of a geometric series is a useful tool for finding the sum of a sequence of numbers where each term is generated by multiplying the previous term by a fixed constant (the common ratio, $�$). The formula depends on the number of terms in the series, the first term (${�}_1$), and the common ratio ($�$). Here's the formula for the sum of a geometric series:

Sum of a Geometric Series (${�}_{�}$):

${�}_{�}=\frac{{�}_1(1-{�}^{�})}{1-�}$

Where:

• ${�}_{�}$ is the sum of the first $�$ terms of the series.
• $�$ is the number of terms.
• ${�}_1$ is the first term.
• $�$ is the common ratio.

Let's use this formula in a couple of examples:

Example 1: Find the sum of the first 5 terms of the geometric series 2, 6, 18, 54, ...

• Number of terms ($�$): 5
• First term (${�}_1$): 2
• Common ratio ($�$): 3

Using the formula:

${�}_5=\frac{2(1-3^5)}{1-3}=\frac{2(1-243)}{-2}=\frac{2(-242)}{-2}=242$

So, the sum of the first 5 terms of the series is 242.

Example 2: Find the sum of the first 8 terms of the geometric series 1, 4, 16, 64, ...

• Number of terms ($�$): 8
• First term (${�}_1$): 1
• Common ratio ($�$): 4

Using the formula:

${�}_8=\frac{1(1-4^8)}{1-4}=\frac{1(1-65536)}{-3}=\frac{1(-65535)}{-3}=\frac{65535}{3}=21845$

So, the sum of the first 8 terms of the series is 21845.

Example 3: Find the sum of the first 10 terms of the geometric series -3, -6, -12, -24, ...

• Number of terms ($�$): 10
• First term (${�}_1$): -3
• Common ratio ($�$): 2

Using the formula:

${�}_{10}=\frac{-3(1-2^{10})}{1-2}=\frac{-3(1-1024)}{-1}=\frac{-3(-1023)}{-1}=\frac{3069}{1}=3069$

So, the sum of the first 10 terms of the series is 3069.

The formula for the sum of a geometric series simplifies the process of finding the sum when you know the first term, common ratio, and the number of terms. It's a valuable tool for various mathematical and real-world applications, including finance, exponential growth, and engineering.

The formula for the sum of the first $�$ terms of a geometric series, often denoted as ${�}_{�}$, is as follows:

Sum of a Geometric Series (${�}_{�}$):

${�}_{�}=\frac{{�}_1(1-{�}^{�})}{1-�}$

Where:

• ${�}_{�}$ is the sum of the first $�$ terms of the series.
• $�$ is the number of terms.
• ${�}_1$ is the first term of the series.
• $�$ is the common ratio between terms.

This formula allows you to find the sum of a geometric series when you know the number of terms, the first term, and the common ratio.

Here's a breakdown of how to use the formula:

1. ${�}_1$ is the first term of the geometric series.
2. $�$ is the common ratio between terms.
3. $�$ is the number of terms you want to sum.
4. Plug these values into the formula to calculate ${�}_{�}$.

The formula is particularly useful in various mathematical and real-world applications, such as compound interest calculations, exponential growth or decay problems, and probability problems involving geometric distributions.

The formula for finding the sum of an infinite geometric series is a useful tool for calculating the sum when the series extends indefinitely. To use this formula, you need to know the first term (${�}_1$) and the common ratio ($�$) between terms in the series. Here's the formula for the sum of an infinite geometric series:

Sum of an Infinite Geometric Series (${�}_{\mathrm{\infty }}$):

${�}_{\mathrm{\infty }}=\frac{{�}_1}{1-�}$

Where:

• ${�}_{\mathrm{\infty }}$ is the sum of the infinite geometric series.
• ${�}_1$ is the first term of the series.
• $�$ is the common ratio between terms.

Let's use this formula in an example:

Example: Find the sum of the infinite geometric series $4+2+1+\frac{1}{2}+\dots$.

In this series:

• First term (${�}_1$): 4
• Common ratio ($�$): $\frac{1}{2}$

Using the formula:

${�}_{\mathrm{\infty }}=\frac{4}{1-\frac{1}{2}}=\frac{4}{\frac{1}{2}}=4\cdot 2=8$

So, the sum of the infinite geometric series is 8.

This formula is especially valuable when dealing with series that continue indefinitely and where the common ratio is within the range -1 < r < 1, ensuring that the series converges to a finite sum.

To determine whether the sum of an infinite geometric series is defined (i.e., converges to a finite value) or not (i.e., diverges to infinity or negative infinity), you need to consider the common ratio ($�$) between terms. The convergence or divergence of an infinite geometric series depends on the value of (r).

Here are the conditions for the sum of an infinite geometric series to be defined:

1. Convergence Condition: The series converges (the sum is defined) if and only if the absolute value of the common ratio ($\mathrm{\mid }�\mathrm{\mid }$) is less than 1.

   • If $\mathrm{\mid }�\mathrm{\mid }<1$, the series converges to a finite value.
   • If $\mathrm{\mid }�\mathrm{\mid }\ge 1$, the series diverges (the sum is not defined).

2. Divergence Condition: If $\mathrm{\mid }�\mathrm{\mid }\ge 1$, the series diverges.

In summary:

• If $\mathrm{\mid }�\mathrm{\mid }<1$, the infinite geometric series converges and has a finite sum.
• If $\mathrm{\mid }�\mathrm{\mid }\ge 1$, the infinite geometric series diverges and does not have a finite sum.

Let's look at a few examples to illustrate these concepts:

Example 1: Determine whether the infinite geometric series $2+1+\frac{1}{2}+\frac{1}{4}+\dots$ converges or diverges.

In this series:

• First term (${�}_1$): 2
• Common ratio ($�$): $\frac{1}{2}$

Since $\mathrm{\mid }�\mathrm{\mid }=\frac{1}{2}<1$, the series converges. The sum of this series is defined, and you can use the formula for the sum of an infinite geometric series to find it.

Example 2: Determine whether the infinite geometric series $3-6+12-24+\dots$ converges or diverges.

In this series:

• First term (${�}_1$): 3
• Common ratio ($�$): -2

Since $\mathrm{\mid }�\mathrm{\mid }=2\ge 1$, the series diverges. The sum of this series is not defined.

Example 3: Determine whether the infinite geometric series $7,-7,7,-7,\dots$ converges or diverges.

In this series:

• First term (${�}_1$): 7
• Common ratio ($�$): -1

Since $\mathrm{\mid }�\mathrm{\mid }=1$, the series does not meet the convergence condition ($\mathrm{\mid }�\mathrm{\mid }<1$. Therefore, it diverges, and the sum is not defined.

In practice, when working with geometric series, check the absolute value of the common ratio. If $\mathrm{\mid }�\mathrm{\mid }<1$, you can find the sum using the formula for the sum of an infinite geometric series. If $\mathrm{\mid }�\mathrm{\mid }\ge 1$, the series diverges, and you can't find a finite sum.

Finding the sum of an infinite series can be done using various methods, including formulas for specific types of series and techniques such as limits. Let's explore how to find the sum of infinite series with some examples:

Example 1: Find the sum of the infinite geometric series $2+1+\frac{1}{2}+\frac{1}{4}+\dots$.

In this series, the first term (${�}_1$) is 2, and the common ratio ($�$) is $\frac{1}{2}$. To find the sum, we use the formula for the sum of an infinite geometric series:

${�}_{\mathrm{\infty }}=\frac{{�}_1}{1-�}$

Substituting the values:

${�}_{\mathrm{\infty }}=\frac{2}{1-\frac{1}{2}}=\frac{2}{\frac{1}{2}}=4$

So, the sum of the infinite geometric series is 4.

Example 2: Find the sum of the infinite series $1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\dots$.

This is a geometric series with the first term (${�}_1$) equal to 1 and a common ratio ($�$) of $\frac{1}{3}$. Using the formula:

${�}_{\mathrm{\infty }}=\frac{{�}_1}{1-�}=\frac{1}{1-\frac{1}{3}}=\frac{1}{\frac{2}{3}}=\frac{3}{2}$

The sum of the infinite series is $\frac{3}{2}$.

Example 3: Find the sum of the infinite series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots$.

This is an alternating series where each term alternates in sign. To find the sum, you can use the formula for the sum of an infinite alternating series:

${�}_{\mathrm{\infty }}=\frac{{�}_1}{1-�}$

In this series, ${�}_1$ is 1, and $�$ is $-\frac{1}{2}$:

${�}_{\mathrm{\infty }}=\frac{1}{1-(-\frac{1}{2})}=\frac{1}{\frac{3}{2}}=\frac{2}{3}$

So, the sum of the infinite alternating series is $\frac{2}{3}$.

These are examples of geometric and alternating series for which we have specific formulas to find the sums. For more complex series or those without a simple formula, finding the sum might require more advanced mathematical techniques, such as power series, Taylor series, or limits.

The formula for finding the sum of an infinite geometric series is as follows:

Sum of an Infinite Geometric Series (${�}_{\mathrm{\infty }}$):

${�}_{\mathrm{\infty }}=\frac{{�}_1}{1-�}$

Where:

• ${�}_{\mathrm{\infty }}$ is the sum of the infinite geometric series.
• ${�}_1$ is the first term of the series.
• $�$ is the common ratio between terms.

This formula provides a concise way to calculate the sum of a geometric series that continues indefinitely, provided that the common ratio ($�$) falls within the range -1 < r < 1. If $\mathrm{\mid }�\mathrm{\mid }$ is not within this range, the series diverges, and the sum is not defined.

To use the formula:

1. Identify the first term (${�}_1$) of the geometric series.
2. Determine the common ratio ($�$) between terms.
3. Plug these values into the formula to find ${�}_{\mathrm{\infty }}$.

This formula is particularly useful in various mathematical and real-world applications, including finance, exponential growth or decay, and probability problems involving geometric distributions.

An annuity is a series of equal payments made at regular intervals. Annuity problems often involve the calculation of future values, present values, and periodic payments. Here are a couple of examples to illustrate how to solve annuity problems:

Example 1: Future Value of an Ordinary Annuity

Suppose you invest $1,000 at the end of each year for 5 years in an account with an annual interest rate of 6%. What will be the future value of the annuity?

Solution: The future value of an ordinary annuity can be calculated using the formula:

$��=�\times \frac{(1+�{)}^{�}-1}{�}$

Where:

• $��$ is the future value of the annuity,
• $�$ is the periodic payment,
• $�$ is the interest rate per period, and
• $�$ is the number of periods.

For this example:

• P = $1,000,
• $�=0.06$ (6% interest rate), and
• $�=5$ (5 years).

$��=1000\times \frac{(1+0.06{)}^5-1}{0.06}\approx 5593.83$

So, the future value of the annuity after 5 years will be approximately $5,593.83.

Example 2: Present Value of an Annuity Due

Suppose you want to know how much money you need to deposit today in order to receive $800 at the beginning of each month for the next 10 years. The interest rate is 8% per year, compounded monthly.

Solution: The present value of an annuity due can be calculated using the formula:

$��=\frac{�\times (1-(1+�{)}^{-�})}{�}\times (1+�)$

For this example:

• P = $800,
• $�=\frac{0.08}{12}$ (monthly interest rate), and
• $�=10\times 12$ (10 years with monthly compounding).

$��=\frac{800\times (1-(1+\frac{0.08}{12}{)}^{-120})}{\frac{0.08}{12}}\times (1+\frac{0.08}{12})\approx 69,867.50$

So, you would need to deposit approximately $69,867.50 today to receive $800 at the beginning of each month for the next 10 years.

These examples illustrate the application of annuity formulas in calculating the future value and present value of annuities. Keep in mind that variations of annuity problems may involve different compounding periods, payment frequencies, and interest rates.

1. 

What is an nth$�\text{th}$ partial sum?

2. 

What is the difference between an arithmetic sequence and an arithmetic series?

3. 

What is a geometric series?

4. 

How is finding the sum of an infinite geometric series different from finding the nth$�\text{th}$ partial sum?

5. 

What is an annuity?

1. What is an nth partial sum?
   An nth partial sum, denoted as ${�}_{�}$, represents the sum of the first $�$ terms of a sequence or a series. In the context of sequences or series, the nth partial sum serves as a way to find the sum of a finite portion of the entire sequence or series. The value of $�$ determines how many terms are added together.

2. What is the difference between an arithmetic sequence and an arithmetic series?
   An arithmetic sequence is a list of numbers in a specific order where each term is obtained by adding a fixed constant (the common difference) to the previous term. An example of an arithmetic sequence is 2, 4, 6, 8, 10, where the common difference is 2.

An arithmetic series, on the other hand, is the sum of the terms in an arithmetic sequence. It represents the result of adding all the terms in the sequence. For the example sequence mentioned above, the arithmetic series would be 2 + 4 + 6 + 8 + 10.

3. What is a geometric series?
   A geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed constant (the common ratio). In a geometric series, the ratio between any two consecutive terms remains constant. A simple example of a geometric series is $1,2,4,8,16,\dots$, where the common ratio is 2.

4. How is finding the sum of an infinite geometric series different from finding the nth partial sum?
   Finding the sum of an infinite geometric series involves determining the sum when the series extends indefinitely. This is done using a specific formula for an infinite geometric series, which depends on the first term, the common ratio, and the condition that the common ratio must have an absolute value less than 1. It results in a finite value.

Finding the nth partial sum, on the other hand, is about calculating the sum of the first $�$ terms of a geometric series. This is a finite value, representing the sum of a specific, finite portion of the series. The formula for the nth partial sum is different from that of an infinite geometric series.

5. What is an annuity?
   An annuity is a financial arrangement in which a series of equal payments or receipts is made at regular intervals. Annuities are commonly used in financial planning and investments, particularly in retirement planning and insurance. There are various types of annuities, including:

• Fixed Annuity: Provides regular, fixed payments for a predetermined period.
• Variable Annuity: The payments can vary based on the performance of underlying investments.
• Immediate Annuity: Payments begin immediately after a lump-sum payment.
• Deferred Annuity: Payments start at a future date.
• Annuity Due: Payments are made at the beginning of each period.
• Ordinary Annuity: Payments are made at the end of each period.

Annuities can be used for income distribution, savings, and investment purposes. The amount of each payment, the frequency of payments, and the interest rates involved determine the value and characteristics of an annuity.