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    Sequences and Series

    Math

    Study Guide
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      • Sequences and Series
      • General Sequences and Series
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      • Arithmetic Sequences
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      • Geometric Sequences
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    Terms and Formulae

    Terms and Formulae

      Terms

    • Arithmetic Sequence

      A sequence in which each term is a constant amount greater or less than the previous term. In this type of sequence, an+1 = an + d, where d is a constant.

    • Common Ratio

      In a geometric sequence, the ratio r between each term and the previous term.

    • Convergent Series

      A series whose limit as n→∞ is a real number.

    • Divergent Series

      A series whose limit as n→∞ is either ∞ or - ∞.

    • Explicit Formula

      A formula for the nth term of a sequence of the form an = some function of n.

    • Finite Sequence

      A sequence which is defined only for positive integers less than or equal to a certain given integer.

    • Finite Series

      A series which is defined only for positive integers less than or equal to a certain given integer.

    • Geometric Sequence

      A sequence in which the ratio between each term and the previous term is a constant ratio.

    • Index of Summation

      The variable in the subscript of Σ. For an, i is the index of summation.

    • Infinite Sequence

      A sequence which is defined for all positive integers.

    • Infinite Series

      A series which is defined for all positive integers.

    • Recursive Sequence

      A sequence in which a general term is defined as a function of one or more of the preceding terms. A sequence is typically defined recursively by giving the first term, and the formula for any term an+1 after the first term.

    • Sequence

      A function which is defined for the positive integers.

    • Series

      A sequence in which the terms are summed, not just listed.

    • Summation Notation

      an = a1 + a2 + a3 + a4 + ... + an. The symbol Σ and its subscript and superscript are the components of summation notation.

    • Term

      An element in the range of a sequence. A sequence is rarely represented by ordered pairs, but instead by a list of its terms.

    • Formulae

       
      Limit of an Infinite Geometric Series For a geometric sequence an = a1rn-1, where -1 < r < 1, the limit of the infinite geometric series a1rn-1 = . This is the same as the sum of the infinite geometric sequence an = a1rn-1.
       
      Sum of a Finite Arithmetic Sequence The sum of the first n terms of the arithmetic sequence is Sn = n() or Sn = na1 + (dn - d ), where d is the difference between each term.
       
      Sum of a Finite Geometric Sequence For a geometric sequence an = a1rn-1, the sum of the first n terms is Sn = a1().
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