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

Semiformal Definition of a "Series":
A series sum (k=a..b) 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:
  sum (k=a..b) an = AA + AA+1 + AA+2 + ... + AB-1 + AB

Summation Arithmetic:
sum (k=a..b) c an = c sum (k=a..b) an (constant c)

sum (k=a..b) an + sum (k=a..b) bn = sum (k=a..b) an + bn

sum (k=a..b) an - sum (k=a..b) bn = sum (k=a..b) an - bn

Summation Identities on the Bounds:
b
sum an
n=a
   c
+ sum an
  n=b+1
   c
= sum an
  n = a

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

b
sum an
n=a
   g(b)
= sum ag -1(c)
   n=g(a)

  
 
  

 
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