Series and Their Notations: Fresh Take

  • Use summation notation to write a sum for a series
  • Use the formula for the sum of the first [latex]n[/latex] terms of an arithmetic series
  • Use the formula for the sum of the first [latex]n[/latex] terms of a geometric series
  • Use the formula to accurately find the sum of an infinite geometric series
  • Solve annuity problems by applying concepts of regular series additions

Series and Summation Notation

The Main Idea

  • Series Definition:
    • A series is the sum of the terms in a sequence
    • It represents the total of a sequence of numbers
  • Summation Notation (Sigma Notation):
    • Uses the Greek capital letter sigma ([latex]\Sigma[/latex])
    • General form: [latex]\sum_{k=1}^{n} a_k[/latex]
    • Represents the sum of [latex]a_k[/latex] from [latex]k=1[/latex] to [latex]k=n[/latex]
  • Components of Summation Notation:
    • Index of summation: The variable used (e.g., [latex]k[/latex])
    • Lower limit of summation: The starting value for the index
    • Upper limit of summation: The ending value for the index
    • Explicit formula: The expression to the right of [latex]\Sigma[/latex]
  • Evaluating a Series:
    • Substitute each value of the index from lower to upper limit into the formula
    • Add all resulting terms
Evaluate [latex]\sum\limits _{k=2}^{5}\left(3k - 1\right)[/latex].

You can view the transcript for “Ex 1: Find a Sum Written in Summation / Sigma Notation” here (opens in new window).

Arithmetic Series

The Main Idea

  • Definition of Arithmetic Series:
    • Sum of terms in an arithmetic sequence
    • Arithmetic sequence: difference between consecutive terms is constant
  • Formula for Partial Sum: [latex]S_n = \frac{n}{2}(a_1 + a_n)[/latex]
    • [latex]S_n[/latex]: Sum of first [latex]n[/latex] terms
    • [latex]n[/latex]: Number of terms
    • [latex]a_1[/latex]: First term
    • [latex]a_n[/latex]: [latex]n[/latex]th term
  • Finding [latex]a_n[/latex] in Arithmetic Sequence: [latex]a_n = a_1 + (n - 1)d[/latex]
    • [latex]d[/latex]: Common difference
Find the partial sum of each arithmetic series.

  1. [latex]5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32[/latex]
  2. [latex]20 + 15 + 10 + \dots + -50[/latex]
  3. [latex]\sum\limits _{k=1}^{12}3k - 8[/latex]

Use the formula to find the partial sum of each arithmetic series.

  1.  [latex]1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.4[/latex]

  2.  [latex]12+21+29\dots + 69[/latex]

  3. [latex]\sum\limits _{k=1}^{10}5 - 6k[/latex]

You can view the transcript for “Introduction to Arithmetic Series” here (opens in new window).

Geometric Series

The Main Idea

  • Definition of Geometric Series:
    • Sum of terms in a geometric sequence
    • Geometric sequence: ratio between consecutive terms is constant
  • Formula for Sum of First n Terms: [latex]S_n = \frac{a_1(1-r^n)}{1-r}, \text{ where } r \neq 1[/latex]
    • [latex]S_n[/latex]: Sum of first [latex]n[/latex] terms
    • [latex]a_1[/latex]: First term
    • [latex]r[/latex]: Common ratio
    • [latex]n[/latex]: Number of terms
Use the formula to find the indicated partial sum of each geometric series.
[latex]{S}_{20}[/latex] for the series [latex]1\text{,}000 + 500 + 250 + \dots[/latex]

Use the formula to determine the sum [latex]\sum\limits _{k=1}^{8}{3}^{k}[/latex]

You can view the transcript for “Geometric Series” here (opens in new window).

Infinite Geometric Series

The Main Idea

  • Definition of Infinite Series:
    • Sum of the terms of an infinite sequence
    • Example: [latex]2 + 4 + 6 + 8 + \cdots = \sum_{k=1}^{\infty} 2k[/latex]
  • Convergence and Divergence:
    • Convergent: Sum approaches a finite value
    • Divergent: Sum is not defined (increases without bound)
  • Convergence of Infinite Geometric Series:
    • Converges when [latex]-1 < r < 1[/latex]
    • [latex]r[/latex]: Common ratio of the geometric sequence
  • Formula for Sum of Infinite Geometric Series: [latex]S_{\infty} = \frac{a_1}{1-r}, \text{ where } |r| < 1[/latex]
    • [latex]S_{\infty}[/latex]: Sum of infinite terms
    • [latex]a_1[/latex]: First term
    • [latex]r[/latex]: Common ratio
Determine whether the sum of the infinite series is defined.

  1. [latex]\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{9}{8}+\cdots[/latex]
  2. [latex]24+(-12)+6+(-3)+\dots[/latex]
  3. [latex]\sum\limits _{k=1}^{\infty} 15\cdot(-0.3)^k[/latex]

Finding Sums of Infinite Series

The Main Idea

  • Formula for Sum of Infinite Geometric Series: [latex]S = \frac{a_1}{1-r}, \text{ where } |r| < 1[/latex]
    • [latex]S[/latex]: Sum of infinite terms
    • [latex]a_1[/latex]: First term
    • [latex]r[/latex]: Common ratio
  • Derivation from Finite Series:
    • Finite series sum: [latex]S_n = \frac{a_1(1-r^n)}{1-r}[/latex]
    • As [latex]n \to \infty[/latex], [latex]r^n \to 0[/latex] when [latex]|r| < 1[/latex]
  • Convergence Condition:
    • Series converges when [latex]-1 < r < 1[/latex]
    • [latex]r^n[/latex] approaches [latex]0[/latex] as [latex]n[/latex] increases
  • Applications:
    • Converting repeating decimals to fractions
    • Solving real-world problems involving infinite processes
Find the sum if it exists.

  1. [latex]2+\dfrac{2}{3}+\dfrac{2}{9}+\dots[/latex]
  2. [latex]\sum\limits _{k=1}^{\infty}{0.76k+1}[/latex]
  3. [latex]\sum\limits _{k=1}^{\infty}\left(-\dfrac{3}{8}\right)^k[/latex]