Dave's Math Tables: Series Properties ![]() |
(Math | Calculus | Expansions | Series | Properties) |
Semi-Formal Definition of a "Series":
A series an is the indicated sum
of all values of an when n is set to each integer from a to b inclusive; namely, the indicated sum of the values aa + aa+1 + aa+2 + ... + ab-1 + ab.
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:
an = aa + aa+1 + aa+2 + ... + ab-1 + ab
Summation Arithmetic:
c an = c
an (constant c)
an +
bn =
an + bn
an -
bn =
an - bn
Summation Identities on the Bounds:
b![]() n=a |
c + ![]() n=b+1 | c = ![]() n = a |
| | (similar relations exist for subtraction and division as generalized below for any operation g) | |
b![]() n=a |
g(b) = ![]() n=g(a) |