Series Properties

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Series Properties

(Math | Calculus | Series Expansions | Properties)

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

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

sum (k=a..b) a_n + b_n = a_n + b_n

sum (k=a..b) a_n - b_n = a_n - b_n

Summation Identities on the Bounds:

b
sum a_n
n=a
c
+ sum a_n
n=b+1
c
= sum a_n
n = a

b
sum a_n
n=a
   b-c
= sum a_n+c
   n=a-c

b
sum a_n
n=a
   b/c
= sum a_nc
   n=a/c

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(similar relations exist for subtraction and division as generalized below for any operation g)
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b
sum a_n
n=a
g(b)
= sum a_{g ^-1(c)}
n=g(a)