Semiformal 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
 an
n=a
|
c
+
an
n=b+1
|
c
=
an
n = a
|
b
an
n=a
|
b-c
=
an+c
n=a-c
|
b
an
n=a
|
b/c
=
anc
n=a/c
| |
|
(similar relations exist
for subtraction and division as generalized below for any operation
g)
| |
b
an
n=a
|
g(b)
= ag
-1(c)
n=g(a)
|
|
