3.1 Summation Formulas and Properties

General
 while and for loops
 sequence a1, a2, a3, ..., an
 finite sum a1 + a2 + ... + an
 infinite sum
which is the same as
 diverges  limit does not exists
 converges  limit exists
 absolutely converget series
 if
converges
 then
also converges

Linearity
 for constant c and sequences a1, a2, ..., an and b1, b2, ..., bn
 also for infinite series
 useful to manipulate asymptotic notation.
 ex:

Arithmetic Series
 arithmetic series:
 has value of:
= (n^{2})

Geometric Series
 geometric or exponential series
 for real x 1
 has value of:
 infinite decreasing geometric series
 infinite summation and x<1
 has values of:

Harmonic Series
 nth Harmonic Number for positive integers n:

Integrating and Differentiating Series
 additional formulas can be obtained by integrating and differentiating
 ex:
 differentiate both sides of
 and then multiply by x, you get

Telescoping Series
 for any series a0, a1, a2, ..., an:
 because each term is added and substracted exactly once
 similarly
 example:
we can write
we get

Products
 finite product a1 · a2 ·
a3 · ... · an
 if n = 0 value is 1
 product to summation identity: