Summation formulas and properties

Converges or diverges based on P_n.

Note: order matters except when

 Linearity:

n å k=1

cak + cbk

=

n å k=1

cak  +

n å cbk k=1

For example:

n å i = 1

Q ( f ( i ))

=  Q (

n å f ( i ) ) i = 1

Some Series:

n å i = 1   i   =

1 + 2 + 3 + ... + n  = n ( n + 1 ) / 2 =  Q ( n2 )   ( arithmetic series )

n å i = 0   xi   =

1 + x  + ... + xn  = ( xn+1 - 1 ) / (x - 1)             (geometric series )

if | x | < 1

¥ å i =0

xi =

lim n ® ¥

n å i =0

xi

=

lim n ® ¥

( xn+1 - 1 ) / (x - 1) =

1/(1 - x)

n å i =1   1 / i   =

1 + 1/2 + ... + 1/n  =  ln n +   O (1 )            (harmonic series )

The series can be integrated term by term when in convergence interval

Telescoping Series:

(a₁ - a₀ ) + (a₂ - a₁ ) + (a₃ - a₂ ) +  ... + (a_n - a_n-1 )  =  (a_n - a₀)

n-1 å k=1

1 /( k ( k + 1))

=

n-1 å k=1

(1/k -1/( k + 1))

=

1/1 - 1/(n -1 +1)

=

1-1/n

Products:

n Õ i = 1   ai   =

a1 x a2 x a3 x ...x an

When n = 0  Õ a_i  = 1, by definition

 

lg

n Õ ai i = 1

=

n å  lg ai i = 1