Further series

Further Series Revision Content

Sequence and Series Basics

  • Understanding notation: ‘u’ denotes terms in a sequence. For example, ‘u1’ refers to the first term, ‘u2’ to the second term, etc.
  • A sequence is a list of numbers in a specific order.
  • A series is the sum of terms in a sequence.
  • Sequences can be finite or infinite. An infinite sequence can approach a finite limit, known as converging, or approach infinity, known as diverging.

Arithmetic Series

  • An arithmetic series is a sequence where each term is a fixed amount larger than the previous term.
  • The formula for the nth term of an arithmetic series is u(n) = a + (n - 1)d, where ‘a’ is the first term and ‘d’ is the common difference.
  • The sum of the first n terms in an arithmetic series can be found using S(n) = n/2 [2a + (n - 1)d].

Geometric Series

  • A geometric series is a sequence where each term is multiplied by a fixed amount, known as the common ratio, to get the next term.
  • The formula for the nth term of a geometric series is u(n) = ar^(n - 1), where ‘a’ is the first term and ‘r’ is the common ratio.
  • The sum of the first n terms in a geometric series can be found using S(n) = a (1 - r^n) / (1 - r) if ‘r’ is not equal to 1.

Convergence of Series

  • A series is said to converge if the sum of the terms approaches a finite number as the number of terms increases.
  • For a geometric series to converge, the absolute value of the common ratio must be less than 1. This can be represented as ** r < 1**.

Binomial Theorem

  • The binomial theorem allows for the expansion of powers of a binomial (a + b)^n.
  • The formula is (a + b)^n = Σ [nCr a^(n - r) b^r] from r = 0 to n.
  • The coefficients of the terms in the expansion follow a pattern known as Pascal’s triangle.

Maclaurin Series

  • The Maclaurin series is a special case of the Taylor series centered at zero.
  • It is used to approximate functions having the form f(x) = Σ [f^n(0) x^n / n!] from n = 0 to infinity.
  • The Maclaurin series for common functions like e^x, sin(x), cos(x), and ln(1 + x) should be memorised.

Sigma Notation

  • Sigma notation is a concise way to write the sum of a series.
  • The general form is Σ u(n) from ‘n = a’ to ‘b’, where ‘a’ is the starting term, ‘b’ is the ending term, and ‘u(n)’ is the term expression.
  • Properties such as splitting, shifting, reversing a sum, are relevant for manipulation of sigma notation.