Series and Sequences - Exam Notes

Arithmetic Series

  • Sum of the first n terms: SnS_n

  • Formula: S<em>n=n2[2a</em>1+(n−1)d]S<em>n = \frac{n}{2}[2a</em>1 + (n-1)d] or S<em>n=n2(a</em>1+an)S<em>n = \frac{n}{2}(a</em>1 + a_n) when the first and last terms are known.

  • Partial Sum: The sum of a finite number of terms.

Geometric Series

  • Sum of the first n terms: SnS_n

  • Finite Geometric Series: S<em>n=a</em>1(1−rn)(1−r)S<em>n = a</em>1 \frac{(1 - r^n)}{(1 - r)}, where r≠1r \neq 1

Infinite Geometric Series

  • Divergent: If |r| > 1, the sum is too large to be evaluated.

  • Convergent: If |r| < 1, the series approaches a particular number as terms increase.

  • Formula: S=a1(1−r)S = \frac{a_1}{(1 - r)}

Sigma Notation

  • Represents the sum of a series.

  • Form: ∑x=1n(expression)\sum_{x=1}^{n} (expression)

  • nn: stopping point

  • xx: variable

Pascal's Triangle

  • Diagonals: 1st diagonal is all 1s, 2nd is counting numbers, 3rd is triangular numbers, 4th is tetrahedral numbers.

  • Symmetrical.

  • Horizontal Sums: Powers of 2.

  • Finding nth element of rth row: (nr)=n!(n−r)!r!\binom{n}{r} = \frac{n!}{(n-r)! r!}

Binomial Theorem

  • Expansion of (a+b)n(a + b)^n

  • There are n+1n + 1 terms.

  • The exponent of aa decreases, and the exponent of bb increases in each term.

  • Sum of exponents in each term = nn

  • Formula for finding a specific term: (nr)an−rbr\binom{n}{r} a^{n-r} b^r

  • To find the coefficient of a specific term, use the binomial coefficient formula.