1.4 Sequences and Series

• What is a sequence in mathematics?

  Ordered list of numbers
• A series is the sum of the terms of a sequence
• The notation for a sequence is ${a_{n}}$ or $a_{1}, a_{2}, a_{3}, \dots$
• What does the notation $\sum_{n = 1}^{k} a_{n}$ represent?


  Finite series sum
• In an arithmetic sequence, the difference between consecutive terms is called the common difference.
• The nth term of an arithmetic sequence is given by a_{n} = a_{1} + (n - 1)d</latex>
• What is the formula for the sum of the first n terms of an arithmetic series using $a_{1}$ and $a_{n}$?


  $S_{n} =$$\frac{n}{2}(a_{1} +$$a_{n})$
• In an arithmetic sequence, the constant difference between terms is called the common difference.
• How do you find the 5th term of an arithmetic sequence with $a_{1} =$$3$ and d =2</latex>?


  $a_{5} =$$11$
• In a geometric sequence, each term is multiplied by a constant called the common ratio.
• What is the formula for the nth term of a geometric sequence?

  a_{n} = a_{1} \times r^{(n - 1)}</latex>
• The sum of the first n terms of a geometric series is given by $S_{n} =$$\frac{a_{1}(1 - r^{n})}{1 - r}$
• Under what condition does an infinite geometric series converge?

  $|r| < 1$
• Convergence describes a sequence or series whose terms approach a fixed value.
• What does it mean for a series to diverge?

  Terms do not approach a fixed value
• The sequence ${1 / n}$ converges to 0.
• What is the sum of the infinite geometric series $\sum_{n = 1}^{\infty} \frac{1}{2^{n}}$?


  $S_{\infty} =$$1$
• What do the terms convergence and divergence describe in mathematics?

  Behavior of sequences and series
• A sequence or series converges if its terms or partial sums approach a fixed value
• A series diverges if its terms approach infinity.
• What is used to determine the convergence of a sequence or series?

  Limit as n approaches infinity
• The sequence ${1 / n}$ converges to 0
• To what value does the sequence ${n}$ diverge?


  $\infty$
• The series $\sum_{n = 1}^{\infty} \frac{1}{2^{n}}$ converges to 1
• To what value does the series $\sum_{n = 1}^{\infty} n$ diverge?


  $\infty$
• Match the term with its definition and example:

  Convergence ↔️ Approach a fixed value ||| $\lim_{n \to \infty} \frac{1}{n} =$$0$

  Divergence ↔️ Do not approach a fixed value ||| $\lim_{n \to \infty} n =$$\infty$
• What is a sequence in mathematics?

  Ordered list of numbers
• A series is the sum of the terms in a sequence.
• What is the notation for a sequence?

  ${a_{n}}$
• The sequence $1, 4, 9, 16, \dots$ is an example of a sequence
• What is an example of a series?

  $1 +$$4 +$$9 +$$16 +$$\dots$
• The sequence $2, 4, 6, 8, \dots$ can be summed to form the series $2 +$$4 +$$6 +$$8 +$$\dots$.
• What is the defining characteristic of an arithmetic sequence?

  Constant difference between terms
• The nth term formula for an arithmetic sequence is $a_{n} =$$a_{1} +$$(n - 1)d$, where $a_{1}$ is the first term and $d$ is the common difference
• What is an arithmetic series?

  Sum of arithmetic sequence
• What is the formula for the nth term of an arithmetic sequence?

  $a_{n} =$$a_{1} +$$(n - 1)d$
• An arithmetic series is the sum of the terms in an arithmetic sequence
• The sum of the first n terms of an arithmetic series is given by $S_{n} =$$\frac{n}{2}(a_{1} +$$a_{n})$.
• What does 'd' represent in the formula for the nth term of an arithmetic sequence?

  Common difference
• The constant difference between consecutive terms in an arithmetic sequence is called the common difference