Comprehensive Notes on Sequences and Series

Definitions of Sequences and Series

• Sequence: A sequence is defined as a function whose domain is the set of positive integers or the set of finite integers $\{1, 2, 3, \dots, n\}$.

• Series: A series represents the sum of the terms of a sequence. If a sequence is finite, the sum of its terms is referred to as the series associated with that sequence. If the sequence contains infinitely many terms, the sum is defined with more precision using calculus.

Comparison of Sequences and Series

• Structural Difference: A sequence is a list of numbers separated by commas, whereas a series is a sum of numbers separated by addition ($+$) or subtraction ($-$) signs.

• Illustration:

  • Sequence: $1, -1, \frac{1}{2}, -\frac{1}{4}$
  • Associated Series: $1 - 1 + \frac{1}{2} - \frac{1}{4} = \frac{1}{4}$

• Notation: The sequence with the $n$-th term $a_n$ is typically denoted by $\{a_n\}$. The associated series results in a total sum $S$, given by:     $S = a_1 + a_2 + a_3 + \dots + a_n$

Specific Examples and Associated Series

To determine the first five terms of a sequence and their associated series, the index $n$ is substituted with values from $1$ to $5$, and the terms are summed to find $S = a_1 + a_2 + a_3 + a_4 + a_5$.

• Sequence $\{2 - n\}$

  • $a_1 = 2 - 1 = 1$
  • $a_2 = 2 - 2 = 0$
  • $a_3 = 2 - 3 = -1$
  • $a_4 = 2 - 4 = -2$
  • $a_5 = 2 - 5 = -3$
  • Associated Series: $S = 1 + 0 - 1 - 2 - 3 = -5$

• Sequence $\{1 + 2n + 3n^2\}$

  • $a_1 = 1 + 2(1) + 3(1)^2 = 6$
  • $a_2 = 1 + 2(2) + 3(2)^2 = 17$
  • $a_3 = 1 + 2(3) + 3(3)^2 = 34$
  • $a_4 = 1 + 2(4) + 3(4)^2 = 57$
  • $a_5 = 1 + 2(5) + 3(5)^2 = 86$
  • Associated Series: $S = 6 + 17 + 34 + 57 + 86 = 200$

• Sequence $\{(-1)^n\}$

  • $a_1 = (-1)^1 = -1$
  • $a_2 = (-1)^2 = 1$
  • $a_3 = (-1)^3 = -1$
  • $a_4 = (-1)^4 = 1$
  • $a_5 = (-1)^5 = -1$
  • Associated Series: $S = -1 + 1 - 1 + 1 - 1 = -1$

• Sequence $\{1 + 2 + 3 + \dots + n\}$

  • $a_1 = 1$
  • $a_2 = 1 + 2 = 3$
  • $a_3 = 1 + 2 + 3 = 6$
  • $a_4 = 1 + 2 + 3 + 4 = 10$
  • $a_5 = 1 + 2 + 3 + 4 + 5 = 15$
  • Associated Series: $S = 1 + 3 + 6 + 10 + 15 = 35$

Arithmetic Sequences and Series

• Arithmetic Sequence: A sequence where each term after the first is found by adding a constant value, known as the common difference ($d$), to the term immediately preceding it.

• General Term ($n$-th term): If $a_1$ is the first term and $d$ is the common difference:     $a_n = a_1 + (n - 1)d$

• Arithmetic Series Formulas: The sum of an arithmetic series with $n$ terms can be calculated using either of the following formulas:     $S_n = \frac{n(a_1 + a_n)}{2}$$S_n = \frac{n[2a_1 + (n - 1)d]}{2}$

Geometric Sequences and Series

• Geometric Sequence: A sequence where each term after the first is found by multiplying the preceding term by a constant value, known as the common ratio ($r$).

• General Term ($n$-th term): If $a_1$ is the first term and $r$ is the common ratio:     $a_n = a_1 \cdot r^{n-1}$

• Geometric Series Formulas: The sum of a geometric series with $n$ terms depends on the value of the common ratio $r$:

  • If $r = 1$:         $S_n = n a_1$
  • If $r \neq 1$:         $S_n = \frac{a_1(1 - r^n)}{1 - r}$

Infinite Geometric Series and Harmonic Sequences

• Infinite Geometric Series: An infinite geometric series of the form $a_1 + a_1r + a_1r^2 + \dots + a_1r^{n-1} + \dots$ has a finite sum if and only if $-1 < r < 1$. The sum is given by:     $S = \frac{a_1}{1 - r}$

• Harmonic Sequence: If a sequence $\{a_n\}$ is an arithmetic sequence, then the sequence where each term is the reciprocal of the arithmetic term ($1/a_n$) is defined as a harmonic sequence.

Identification and Classification Exercises

Sequence (SEQ) vs. Series (SER) Identification:

• (a) $1, 2, 4, 8, \dots$: SEQ
• (b) $2, 8, 10, 18, \dots$: SEQ
• (c) $-1 + 1 - 1 + 1 - 1$: SER
• (d) $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$: SEQ
• (e) $1 + 2 + 2^2 + 2^3 + 2^4$: SER
• (f) $1 + 0.1 + 0.001 + 0.0001$: SER

Sequence Classification (A = Arithmetic, G = Geometric, F = Fibonacci, O = Other):

• (a) $3, 5, 7, 9, 11, \dots$: A ($d = 2$)
• (b) $2, 4, 9, 16, 25, \dots$: O
• (c) $\\frac{1}{4}, \\frac{1}{16}, \\frac{1}{64}, \\frac{1}{256}, \dots$: G ($r = 1/4$)
• (d) $\\frac{1}{2}, \\frac{2}{3}, \\frac{3}{4}, \\frac{4}{5}, \dots$: O
• (e) $\\frac{1}{5}, \\frac{1}{9}, \\frac{1}{13}, \\frac{1}{17}, \\frac{1}{21}, \dots$: O (Harmonic based on Arithmetic $5, 9, 13, 17, 21$)
• (f) $4, 6, 10, 16, 26, \dots$: F (General Fibonacci type; $4+6=10, 6+10=16, 10+16=26$)
• (g) $\sqrt{3}, \sqrt{4}, \sqrt{5}, \sqrt{6}, \dots$: O
• (h) $0.1, 0.01, 0.001, 0.0001, \dots$: G ($r = 0.1$)

Computation and Summation Exercises

Sequence Term Determination:

1. Sequence $\{1 + n - n^2\}$: Determine the first five terms and the series sum.
2. Sequence $\{1 - (-1)^{n+1}\}$: Determine the first five terms and the series sum.
3. Recursive Sequence: Given $a_1 = 3$ and $a_n = 2a_{n-1} + 3$ for $n \geq 2$, find terms and sum.
4. Factorial-style Sequence: $\{1 \cdot 2 \cdot 3 \cdot \dots \cdot n\}$: Find terms and sum.

Series Type and Sum Determination: For these items, if the series is not arithmetic, geometric, or infinite geometric, it is labeled NAGIG. If a sum cannot be calculated, it is labeled NO SUM.

• (a) $4 + 9 + 14 + \dots + 64$: Arithmetic ($d = 5$).
• (b) $81 + 27 + 9 + \dots + \frac{1}{81}$: Geometric ($r = 1/3$).
• (c) $1 + 3 + 6 + 10 + 15 + 21 + \dots + 55$: NAGIG.
• (d) $-10 - 2 + 6 + \dots + 46$: Arithmetic ($d = 8$).
• (e) $10 + 2 + 0.4 + 0.08 + \dots$: Infinite Geometric ($r = 0.2$).
• (f) $\\frac{1}{3} + \\frac{1}{3} + \\frac{1}{3} + \\frac{1}{3} + \dots$: Infinite Geometric ($r = 1$), but sum results in NO SUM because $|r|$ is not less than 1.
• (g) $1 - 0.1 + 0.01 - 0.001 + \dots$: Infinite Geometric ($r = -0.1$).