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

2. 2 Summation notation akak k k0k0 kfkf k 0 - 1 k 0 + 1 k f + 1 k f - 1

3. 3 Summation notation akak k k0k0 kfkf k 0 - 1 k 0 + 1 k f + 1 k f - 1

4. 4 Summation notation

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

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

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

8. 8 Geometric series Ar k k 1... 0234567 Ar 0 Ar 1 Ar 2 Ar 3 Ar 4 Ar 5 Ar 6 Ar 7 r < 1 (zoom-out vertical scale)

9. 9 Geometric series Ar k k 1... 0234567 r < 1 Repeated addition seems to point toward a finite value

10. 10 Ar k k 1... 023 r < 1 Geometric series

11. 11 Geometric sum: Summation notation Ar k k 1... 023 r < 1 Use j := k – 1 to check this last step STOP

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

13. 13 Harmonic series