Basic Concepts of sequences and series

Sequences:

Definition:

  • A sequence is an ordered list of numbers arranged according to a certain rule or pattern.

Types of Sequences:

  1. Arithmetic Sequence:

    • Defined by a common difference (d) between consecutive terms.
    • General form: an=a1+(n1)d where an is the nth term.
  2. Geometric Sequence:

    • Defined by a common ratio (r) between consecutive terms.
    • General form: an=a1r(n1) where an is the nth term.
  3. Harmonic Sequence:

    • Defined by the reciprocals of an arithmetic sequence.
    • General form: an=1a1+(n1)d where an is the nth term.

Properties of Sequences:

  • Convergent vs. Divergent:

    • Convergent sequences approach a specific limit as n approaches infinity.
    • Divergent sequences do not approach a limit and may tend towards positive or negative infinity.
  • Monotonic Sequences:

    • Monotonic sequences either consistently increase (monotonically increasing) or decrease (monotonically decreasing) with each term.

Series:

Definition:

  • A series is the sum of the terms in a sequence.

Types of Series:

  1. Finite Series:

    • The sum of a specific number of terms in a sequence.
    • Example: Sn=a1+a2+a3+...+an
  2. Infinite Series:

    • The sum of an infinite number of terms in a sequence.
    • Example: S=a1+a2+a3+...+an+...

Properties of Series:

  • Convergence of Infinite Series:

    • Convergent series have a finite sum.
    • Divergent series have an infinite sum or no sum at all.
  • Tests for Convergence:

    • Divergence Test: If the terms of the series do not approach zero, the series diverges.
    • Geometric Series Test: Determines convergence for geometric series based on the common ratio.
    • Integral Test: Applies to series with non-negative, decreasing terms and connects the convergence of series to definite integrals.
    • Comparison Test: Compares the given series with a known series to establish convergence or divergence.
  • Summation of Series:

    • Formulas exist for finding the sum of certain types of series, like arithmetic, geometric, and telescoping series.

Common Formulas:

  • Arithmetic Series Sum: Sn=n2(a1+an)
  • Geometric Series Sum: S=a11r (for r<1)

Applications:

  • Mathematics: Used in calculus, discrete mathematics, and number theory.
  • Physics: Applied in modeling various physical phenomena involving continuous change or discrete values.
  • Computer Science: Used in algorithms, data structures, and cryptography.