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:

= (n2)

• 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: