Summation 1 Summation notation Geometric seriesHarmonic series Gauss summation trick (r < 1)

2 Summation notation akak k k0k0 kfkf k k k f + 1 k f - 1

3 Summation notation akak k k0k0 kfkf k k k f + 1 k f - 1

4 Summation notation

Summation 5 Summation notation Geometric seriesHarmonic series Gauss summation trick (r < 1)

6 Gauss summation trick 2 Figure it out: Why N + 1 individuals? Why not N ?

Summation 7 Summation notation Geometric seriesHarmonic series Gauss summation trick (r < 1)

8 Geometric series Ar k k Ar 0 Ar 1 Ar 2 Ar 3 Ar 4 Ar 5 Ar 6 Ar 7 r < 1 (zoom-out vertical scale)

9 Geometric series Ar k k r < 1 Repeated addition seems to point toward a finite value

10 Ar k k r < 1 Geometric series

11 Geometric sum: Summation notation Ar k k r < 1 Use j := k – 1 to check this last step STOP

Summation 12 Summation notation Geometric seriesHarmonic series Gauss summation trick (r < 1)

13 Harmonic series