Renormalization

By Pijar & Fajar

Summation

1

Summation

1

Summation

1

1+2+3+4=?

Summation

1

1+2+3+4=10

Summation

1

1+2+3+4+5=?

Summation

1

1+2+3+4+5+6=?

Summation

1

1+2+3+4+5+6......=?

Summation

1

Grandi's Series

2

Grandi's Series

2

1-1+1-1+1-1......=?

Grandi's Series

2

(1-1)+(1-1)+(1-1)......=?

0

0

0

+

+

+...=0

Grandi's Series

2

(1-1)+(1-1)+(1-1)......=?

0

0

0

+

+

+...=0

1+(-1+1)+(-1+1)......=?

1

0

0

+...=1

+

+

Grandi's Series

2

11-11-1+11-1+1-11-1+1-1+11-1+1-1+1...

=1=0=1=0=1=?

?

Cezaro Summation

3

Cezaro Summation

3

Cezaro Summation

3

1-1+1-1+1....=?

Cezaro Summation

3

1-1+1-1+1....=?

Cezaro Summation

3

11-11-1+11-1+1-11-1+1-1+1(1,0,1,0,1,0.....)

=1=0=1=0=1

Cezaro Summation

3

(1,0,1,0,1,0....)(1)(1,0)(1,0,1)(1,0,1,0)(1,0,1,0,1)(1/1 , 1/2 , 2/3, 2/4, 3/5...)

=1/1=1/2=2/3=2/4=3/5

Cezaro Summation

3

(1/1 , 1/2 , 2/3, 2/4, 3/5...)

=1/2

1-1+1-1+1-1.....=1/2

Cezaro Summation

3

Cezaro Summation

3

Cezaro Summation

3

1-2+3-4+5....=?

Cezaro Summation

3

11-21-2+31-2+3-41-2+3-4+5(1,-1,2,-2,3,-3.....)

=1=-1=2=-2=3

Cezaro Summation

3

(1,-1,2,-2,3,-3....)(1)(1,-1)(1,-1,2)(1,-1,2,-2)(1,-1,2,-2,3)(1/1 , 0, 2/3, 0, 3/5...)

=1/1=0=2/3=0=3/5

Cezaro Summation

3

(1/1 , 0, 2/3, 0, 3/5...)(1/1 , 1/2 , 2/3, 2/4, 3/5...)

(0+1/2)/2 = 1/4

1-2+3-4+5-6.....=1/4

Cezaro Summation

3

(1-1+1-1+1-1...)21-2+3-4+5-6.....1/4

=

=

Riemann Zeta

4

1+2+3+4+5+6......=?

Riemann Zeta

4

=?

Riemann Zeta

4

=-(1/12)

Riemann Zeta

4

1+2+3+4+5...=-(1/12)

Riemann Zeta

4

=

Riemann Zeta

4

1+1/2+1/3+1/4+...=Harmonic Series