9.2 Series and Summation Notation

Series and summation notation are powerful tools for working with sequences of numbers. They allow us to represent and manipulate long lists of numbers in a compact way, making it easier to analyze patterns and calculate sums.

Arithmetic and geometric series are two common types we'll encounter. Arithmetic series add a constant difference between terms, while geometric series multiply by a constant ratio. Understanding these helps us model real-world situations like interest and population growth.

Arithmetic vs Geometric Series

Defining Arithmetic and Geometric Series

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• An arithmetic series is a sequence of numbers such that the difference between the consecutive terms is constant
  • The constant difference is called the "common difference"
• A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number
  • The fixed, non-zero number is called the "common ratio"
• The terms of an arithmetic series increase or decrease by a constant amount (common difference), while the terms of a geometric series increase or decrease by a constant factor (common ratio)

Representing Arithmetic and Geometric Series

• Both arithmetic and geometric series can be represented using recursive or explicit formulas
  • Recursive formulas define each term based on the previous term(s)
    • For an arithmetic series: $a_n = a_{n-1} + d$, where $a_n$ is the nth term and $d$ is the common difference
    • For a geometric series: $a_n = a_{n-1} \cdot r$, where $a_n$ is the nth term and $r$ is the common ratio
  • Explicit formulas define each term based on its position in the series
    • For an arithmetic series: $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the position, and $d$ is the common difference
    • For a geometric series: $a_n = a_1 \cdot r^{n-1}$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the position, and $r$ is the common ratio

Summation Notation for Series

Introducing Summation Notation

• Summation notation, denoted by the Greek letter sigma ($\Sigma$), is a concise way to represent the sum of a series of numbers
• The general form of summation notation is $\Sigma_{i=m}^{n} a_i$, where:
  • $i$ is the index of summation
  • $m$ is the lower limit
  • $n$ is the upper limit
  • $a_i$ is the general term

Components of Summation Notation

• The index of summation ($i$) represents the variable in the general term ($a_i$) that changes as the series progresses
• The lower limit ($m$) and upper limit ($n$) define the range of values that the index of summation takes on
  • The lower limit is the starting value for the index
  • The upper limit is the ending value for the index
• To evaluate a summation, substitute the values of the index from the lower limit to the upper limit into the general term and add the resulting terms

Calculating Series Sums

Sum of an Arithmetic Series

• The sum of an arithmetic series with $n$ terms, first term $a_1$, and common difference $d$ is given by $S_n = \frac{n}{2}[2a_1 + (n-1)d]$
  • $S_n$ represents the sum of the first $n$ terms
  • $a_1$ is the first term of the series
  • $d$ is the common difference between consecutive terms
  • $n$ is the number of terms in the series

Sum of a Geometric Series

• The sum of an infinite geometric series with first term $a$ and common ratio $r$, where $|r| < 1$, is given by $S_{\infty} = \frac{a}{1-r}$
  • $S_{\infty}$ represents the sum of an infinite geometric series
  • $a$ is the first term of the series
  • $r$ is the common ratio between consecutive terms
  • The condition $|r| < 1$ ensures the series converges to a finite sum
• The sum of a finite geometric series with $n$ terms, first term $a$, and common ratio $r$ is given by $S_n = \frac{a(1-r^n)}{1-r}$
  • $S_n$ represents the sum of the first $n$ terms
  • $a$ is the first term of the series
  • $r$ is the common ratio between consecutive terms
  • $n$ is the number of terms in the series

Solving for Series Sums

• When solving for the sum of a series, it is essential to:
  • Identify the type of series (arithmetic or geometric)
  • Determine the number of terms ($n$)
  • Identify the first term ($a_1$ or $a$)
  • Determine the common difference ($d$) for arithmetic series or the common ratio ($r$) for geometric series
• Once these components are identified, substitute the values into the appropriate formula to calculate the sum of the series

Real-World Applications of Series

Modeling with Arithmetic Series

• Arithmetic series can model situations involving linear growth or decline
  • Simple interest: The interest earned each period remains constant (savings accounts, loans)
  • Salary increases: An employee receives a fixed annual raise (cost-of-living adjustments)
  • Depreciation: The value of an asset decreases by a constant amount each year (vehicles, equipment)

Modeling with Geometric Series

• Geometric series can model situations involving exponential growth or decay
  • Compound interest: The interest earned each period is a fixed percentage of the current balance (investments, credit card debt)
  • Population growth: The population increases by a fixed percentage each generation (bacteria, rabbits)
  • Radioactive decay: The amount of a radioactive substance decreases by a fixed percentage each half-life (carbon-14 dating)

Solving Real-World Problems

• When solving real-world problems, follow these steps:
  • Identify the given information and determine the type of series (arithmetic or geometric)
  • Use the appropriate formula to calculate the sum or any missing terms
  • Real-world applications may require finding the number of terms ($n$), first term ($a_1$ or $a$), common difference ($d$), or common ratio ($r$) based on the given information and the context of the problem
  • Interpret the results in the context of the problem and communicate the solution effectively