Dave's Math Tables: Series Properties |

(Math | Calculus | Expansions | Series | Properties) |

**Semi-Formal Definition of a "Series":**

A series a_{n} is the *indicated* sum
of all values of a_{n} when __n__ is set to each integer from __a__ to __b__ inclusive; namely, the indicated sum of the values a_{a} + a_{a+1} + a_{a+2} + ... + a_{b-1} + a_{b}.

**Definition of the "Sum of the Series":**

The "sum of the series" is the *actual result* when all the terms of the series are summed.

**Note the difference:** "1 + 2 + 3" is an example of a "series," but "6" is the actual "sum of the series."

**Algebraic Definition:**

a_{n} = a_{a} + a_{a+1} + a_{a+2} + ... + a_{b-1} + a_{b}

**Summation Arithmetic:**

c a_{n} = c a_{n} *(constant c)*

a_{n} + b_{n} = a_{n} + b_{n}

a_{n} - b_{n} = a_{n} - b_{n}

**Summation Identities on the Bounds:**

b a _{n}n=a |
c + a _{n}n=b+1 | c = a _{n}n = a |

| |(similar relations exist for subtraction and division as generalized below for any operation g)| |

b a _{n}n=a |
g(b) = a _{g -1(c)}n=g(a) |