Sequences and Series

• Sequence
  • Definition: Function whose domain is the set of natural numbers $\mathbb{N}$
  • Representation: $a_1, a_2, a_3, \dots, a_n$
  • Finite vs Infinite Sequence
• Series
  • Definition: Sum of the elements of a sequence
  • Representation: $a_1 + a_2 + a_3 + \dots + a_n = \sum_{k=1}^{n} a_k$
  • Finite vs Infinite Series
• Arithmetic Progression (AP)
  • Definition: Sequence where each term after the first differs by a constant common difference ($d$)
  • General Term ($n$-th term): $a_n = a + (n - 1)d$
  • Sum of $n$ terms ($S_n$): $S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)$
  • Arithmetic Mean (A.M.): $A = \frac{a + b}{2}$ between two numbers $a$ and $b$
• Geometric Progression (GP)
  • Definition: Sequence where each term after the first is obtained by multiplying the previous term by a constant common ratio ($r$)
  • General Term ($n$-th term): $a_n = ar^{n-1}$
  • Sum of $n$ terms ($S_n$):
    • $S_n = \frac{a(r^n - 1)}{r - 1}$ (if $r > 1$)
    • $S_n = \frac{a(1 - r^n)}{1 - r}$ (if $r < 1$)
    • $S_n = na$ (if $r = 1$)
  • Sum of Infinite GP ($S_\infty$): $S_\infty = \frac{a}{1 - r}$ (where $|r| < 1$)
  • Geometric Mean (G.M.): $G = \sqrt{ab}$ between two numbers $a$ and $b$
• Relationship Between Means
  • A.M. $\ge$ G.M. for any two positive real numbers
• Special Series
  • Sum of first $n$ natural numbers: $\sum n = \frac{n(n + 1)}{2}$
  • Sum of squares of first $n$ natural numbers: $\sum n^2 = \frac{n(n + 1)(2n + 1)}{6}$
  • Sum of cubes of first $n$ natural numbers: $\sum n^3 = \left[\frac{n(n + 1)}{2}\right]^2$