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13.4 Series and Their Notations

3 min read•june 24, 2024

Series notation simplifies complex calculations, making it easier to work with large sums. It's a powerful tool for representing arithmetic and geometric sequences, allowing us to compute sums efficiently using formulas tailored to each type.

Series have real-world applications in finance, population growth, and resource management. Understanding how to calculate sums and apply series formulas to practical scenarios is crucial for solving problems in various fields, from economics to biology.

Series Notation and Formulas

Summation notation interpretation

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  • Summation notation uses the Greek letter sigma (∑\sum) to represent the sum of a series of terms
    • written below ∑\sum symbol, upper written above
    • Expression after ∑\sum symbol represents general term of series
  • : ∑i=1n(a+(i−1)d)\sum_{i=1}^{n} (a + (i-1)d)
    • aa first term, dd , nn number of terms
  • : ∑i=0n−1ari\sum_{i=0}^{n-1} ar^i
    • aa first term, rr , nn number of terms

Arithmetic series sum calculation

  • Sum of calculated using formula: Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)
    • SnS_n sum of first nn terms
    • aa first term
    • dd common difference between consecutive terms
    • nn number of terms in series
  • Example: Sum of first 10 terms of arithmetic series with a=2a=2 and d=3d=3
    • S10=102(2(2)+(10−1)3)=5(4+27)=155S_{10} = \frac{10}{2}(2(2) + (10-1)3) = 5(4 + 27) = 155

Geometric series sum computation

  • Sum of calculated using formula: Sn=a(1−rn)1−rS_n = \frac{a(1-r^n)}{1-r}, r≠1r \neq 1
    • SnS_n sum of first nn terms
    • aa first term
    • rr between consecutive terms
    • nn number of terms in series
  • Sum of calculated using formula: S∞=a1−rS_\infty = \frac{a}{1-r}, ∣r∣<1|r| < 1
    • S∞S_\infty sum of
    • aa first term
    • rr common ratio, absolute value must be less than 1 for series to converge
  • Example: Sum of infinite geometric series with a=1a=1 and r=12r=\frac{1}{2}
    • S∞=11−12=112=2S_\infty = \frac{1}{1-\frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2

Sequences and Series

  • A is an ordered list of numbers, often represented by a formula
  • An infinite series is the sum of all terms in an infinite sequence
  • The limit of a sequence or series determines its convergence or divergence
  • A defines each term of a sequence based on previous terms

Applications of Series

Series concepts in real-world scenarios

  • Arithmetic series model situations with constant difference between consecutive terms
    • Regular deposits into savings account
    • Linear growth or decline (population, resources)
  • Geometric series model situations with constant ratio between consecutive terms
    • Population growth or decay (bacteria, radioactive material)
    • (investments, loans)

Series formulas for financial problems

  • : series of equal payments made at regular intervals
    • Present value of annuity calculated using formula: PV=PMT⋅1−1(1+r)nrPV = PMT \cdot \frac{1-\frac{1}{(1+r)^n}}{r}
      • PVPV present value of annuity
      • PMTPMT periodic payment amount
      • rr periodic interest rate (decimal form)
      • nn total number of payments
  • : interest calculated on initial principal and accumulated interest from previous periods
    • Future value of lump sum with compound interest calculated using formula: FV=PV(1+r)nFV = PV(1+r)^n
      • FVFV future value
      • PVPV present value (initial principal)
      • rr periodic interest rate (decimal form)
      • nn number of compounding periods
  • Example: Future value of $1,000 invested at 5% annual interest, compounded quarterly for 10 years
    • r=0.05/4=0.0125r = 0.05/4 = 0.0125 (quarterly interest rate)
    • n=4⋅10=40n = 4 \cdot 10 = 40 (number of quarters in 10 years)
    • FV=1000(1+0.0125)40≈FV = 1000(1+0.0125)^{40} \approx 1,643.62

Key Terms to Review (29)

Annuity: An annuity is a series of equal payments made at regular intervals, such as monthly or yearly, for a specified period of time. It is a financial product that provides a steady stream of income, often used for retirement planning or other long-term financial goals.
Arithmetic sequence: An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
Arithmetic Sequence: An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference, and it allows the sequence to be generated by adding the common difference to each term.
Arithmetic Series: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, and it is a key feature that distinguishes an arithmetic series from other types of sequences and series.
Common difference: The common difference is the constant amount added or subtracted between consecutive terms in an arithmetic sequence. It is denoted by $d$.
Common ratio: The common ratio is the constant factor between consecutive terms of a geometric sequence. It is found by dividing any term by its preceding term.
Common Ratio: The common ratio is a constant value that represents the ratio between consecutive terms in a geometric sequence. It is the multiplicative factor that is used to generate each successive term in the sequence from the previous term.
Compound interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It is commonly used in finance and investments to calculate growth over time.
Compound Interest: Compound interest is the interest earned on interest, where the interest accrued on a principal amount is added to the original amount, and the total then earns interest in the next period. This concept is fundamental to understanding the growth of investments and loans over time.
Convergent Series: A convergent series is a type of infinite series where the sum of the terms approaches a finite limit as the number of terms increases. This means that the series has a well-defined sum, and the partial sums of the series converge to this limit.
Divergent Series: A divergent series is a mathematical series where the sum of the terms does not converge to a finite value. In other words, as more terms are added, the sum of the series continues to grow without bound, diverging from any finite limit.
Finite Arithmetic Series: A finite arithmetic series is a sequence of numbers where the difference between consecutive terms is constant. It is a type of series that has a finite number of terms and follows a specific pattern of addition or subtraction.
Finite Geometric Series: A finite geometric series is a series where each term is a constant multiple of the previous term, and the series has a finite number of terms. These series are useful in various mathematical applications, including finance, physics, and computer science.
Geometric sequence: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence can be written as $a, ar, ar^2, ar^3, \ldots$.
Geometric Sequence: A geometric sequence is a sequence where each term is a constant multiple of the previous term. The ratio between consecutive terms is a fixed number, known as the common ratio, which determines the pattern of the sequence.
Geometric Series: A geometric series is a type of infinite series where each term is a constant multiple of the previous term. It is a sequence of numbers where the ratio between consecutive terms is constant, and it can be used to model various real-world phenomena that exhibit exponential growth or decay.
Infinite Geometric Series: An infinite geometric series is a series where each term is a fixed multiple of the previous term, and the series continues indefinitely. These series are used to model various real-world phenomena and have important applications in mathematics and science.
Infinite Series: An infinite series is a sequence of terms that continues indefinitely, where each term is added to the previous terms to form a sum that approaches a specific value or diverges without bound. This concept is central to the study of series and their notations, as explored in the 13.4 section.
Limit: A limit is a value that a function or sequence approaches as the input variable approaches a certain value. It represents the behavior of a function or sequence as it gets closer and closer to a specific point, without necessarily reaching that point.
Lower limit of summation: The lower limit of summation is the starting index value in a summation notation, often denoted by $i=1$ or another integer. It indicates where the series begins.
N→∞: The term 'n→∞' represents the concept of a variable 'n' approaching infinity, or growing without bound. It is a fundamental concept in the study of series and their notations, as it describes the behavior of a sequence or series as the number of terms approaches an arbitrarily large value.
Nth partial sum: The nth partial sum of a series is the sum of the first n terms of that series. It provides an approximation to the total sum of an infinite series by summing a finite number of terms.
Partial Sum: The partial sum of a sequence or series is the sum of the first n terms of the sequence or series. It represents the accumulation of the terms up to a certain point, providing a snapshot of the overall sum as the sequence or series progresses.
Recursive Formula: A recursive formula is a mathematical expression that defines a sequence or series by relating each term to the previous term(s) in the sequence. It allows for the generation of a sequence by repeatedly applying the same rule or formula to generate the next term based on the preceding term(s).
Sequence: A sequence is an ordered list of elements, such as numbers, letters, or objects, that follow a specific pattern or rule. Sequences are fundamental concepts in mathematics and are extensively studied in various topics, including algebra, calculus, and discrete mathematics.
Sigma (Σ): Sigma (Σ) is a mathematical symbol used to represent the summation or addition of a series of numbers or quantities. It is a fundamental concept in the study of series and sequences, which are central topics in the context of 13.4 Series and Their Notations.
Telescoping series: A telescoping series is a specific type of infinite series where most terms cancel out when the series is expanded, leaving only a few terms that determine the sum. This cancellation often simplifies the calculation of the series' limit as the number of terms approaches infinity, making it easier to analyze convergence and evaluate sums.
Π Symbol: The Π symbol, also known as pi, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends or repeats, and it is widely used in various mathematical and scientific fields, including the study of series and their notations.
Σ Notation: Sigma (Σ) notation is a mathematical shorthand used to represent the sum of a series of terms or values. It is a concise way to express the addition of multiple quantities, particularly in the context of series and sequences.
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