Proof for other series

Proof for Other Series Revision Content

Understanding Series

  • A series is the sum of the terms of a sequence.
  • Finite series have a definite number of terms, while infinite series continue indefinitely.
  • In maths, the term ‘series’ is often used to describe a sum of fractions, or a sum of powers of a variable.

Arithmetic Series

  • An arithmetic series is the sum of an arithmetic sequence.
  • The sum, S, of n terms of an arithmetic series can be found using the formula: S = n/2 [2a + (n - 1)d], where ‘a’ is the first term and ‘d’ is the common difference.

Geometric Series

  • A geometric series is the sum of a geometric sequence.
  • For a finite geometric series, the sum, S, of n terms can be found with the formula: S = a(1 - r^n) / (1 - r), where ‘a’ is the first term and ‘r’ is the common ratio.
  • An infinite geometric series will have a sum if the r < 1. The sum, S, can be found with the formula: S = a / (1 - r).

Maclaurin and Taylor Series

  • The Maclaurin series is a series expansion of a function about the point zero.
  • The Taylor series is the general form of the Maclaurin series, and is a series expansion of a function about any point.
  • These series are especially useful for approximating complex functions.

Convergence of Series

  • The convergence of a series refers to the concept that as the number of terms in the series increases, the series sum approaches a fixed value.
  • For geometric series, the series converges if the common ratio ‘r’ is such that r < 1.
  • Tests for convergence for other series include the Ratio Test, the Root Test, and the Comparison Test.

Proof By Mathematical Induction

  • Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers.
  • It involves two steps: the base case (proving the statement is true for initial value), and the inductive step (assuming the statement true for some k, and proving it is true for k+1).
  • Mathematical induction can be used to prove the formulas for the sum of arithmetic and geometric series, among others.

Understanding Sigma Notation

  • Sigma notation is a concise way of expressing long sums, where ‘n’ is variable.
  • The Greek letter Sigma (Σ) is used to denote the sum of a series.
  • Understanding sigma notation is crucial for dealing with series and their proofs.