Terms | |
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Sequence |
a list of numbers, usually separated by commas (t_{1}, t_{2}, t_{3}, ...) (Technically, a sequence is a function whose domain is the natural numbers.) |

Term |
one of the numbers in a sequence or series |

Arithmetic Sequence |
a sequence in which the difference between successive terms is constant |

Common Difference |
the difference between successive terms in an arithmetic sequence |

Geometric Sequence |
a sequence in which the ratio of successive terms is constant |

Common Ratio |
the ratio of successive terms in a geometric sequence |

Finite Sequence |
a sequence that has a last term |

Infinite Sequence |
a sequence that does not have a last term |

Series |
the sum of a sequence (t_{1} + t_{2} + t_{3} + ...) |

Arithmetic Series |
a series based on an arithmetic sequence (the sum of the terms of an arithmetic sequence) |

Geometric Series |
a series based on a geometric sequence (the sum of the terms of a geometric sequence) |

Finite Series |
a series based on a finite sequence (the sum of the terms of a finite sequence) |

Infinite Series |
a series based on an infinite sequence (the sum of the terms of an infinite sequence) |

Partial Sum |
the sum of a finite number of terms of a series |

Sequence of Partial Sums |
every infinite series has a sequence of partial sums = first term, sum of first two terms, sum of first three terms, sum of first four terms, etc. |

Converges |
If the sum of an infinite series has a limit, the series converges. |

Diverges |
If the sum of an infinite series has no limit, the series diverges. |

Explicit Definition |
tells how to calculate any specified term in a sequence or series |

Implicit Definition |
tells how to construct a sequence or series by building it term-by-term starting with the first term |

Limit |
a number that the terms of a sequence (or partial sums of a series) gets closer to as the number of terms increases |

n approaches infinity |
the number n gets larger and larger (forever) |

limit of some expression equals infinity | the value of the expression gets larger and larger (forever) |

Symbols | |
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n | the number of the current (or next) term in a sequence or series |

t | the value of a term in a sequence or series |

d | the common difference of an arithmetic sequence or series |

r | the common ratio of an geometic sequence or series |

t_{n} |
the value of the current (or next) term in a sequence or series |

t_{n-1} |
the value of the previous term in a sequence or series (the term before the n^{th} term) |

S | the value of a series (the sum of terms) |

S_{n} |
the (partial) sum of the first n terms of a series |

S_{n-1} |
the (partial) sum of the first n - 1 terms of a series (the sum of all the terms before term number n) |

What happens to the value of the expression as the number of terms in its sequence (or series) gets larger and larger? | |

Recursive Definitions | |
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Step 1 | Where does the sequence or series start? What is the first term (t_{1})? |

Step 2 | How do you calculate the next term (t The last term so far is t The second to the last term so far is t |

Formulas | |
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Explicit Definition of an Arithmetic Sequence | t _{n} = t_{1} + (n - 1)d |

Explicit Definition of a Geometric Sequence | t _{n} = t_{1}r^{n-1} |

Sum of a Finite Arithmetic Series | |

Sum of a Finite Geometric Series | |

Limit of a Rational Function | |

Limit of a Power Function | |

Sum of an Infinite Geometric Series |

*last update December 18, 2007 by JL Stanbrough*