Infinity and Beyond 5: Examples

Infinite sums

1. Geometric series: ∑n≥02−n
   20+2−1+2−2+2−3+…=2.
2. Telescoping series: ∑n≥11n(n+1)
   11⋅2+12⋅3+13⋅4+14⋅5+…=1.
3. James Gregory's (or Leibniz) series
   11−13+15−17+…=π4.
4. Euler's series: ∑n≥1n−2
   112+122+132+142+⋯=π26.
5. Euler's series: ∑n≥1(2n−1)−2
   112+132+152+172+⋯=π28.
6. Euler's alternating series: ∑n≥1(−1)n+1n−2
   1−122+132−142+152−162+⋯=π212.
7. Euler's alternating series: ∑n≥1(−1)n+1(2n−1)−3
   1−133+153−173+193−1113+⋯=π232.
8. Alternating Harmonic Series:
   11−12+13−14+…=ln(2).
9. Nilakantha (15th century) I:
   115+4⋅1−135+4⋅3+155+4⋅5−175+4⋅7+…=π16.
10. Nilakantha (15th century) II:
    3+433−3−453−5+473−7−493−9+…=π.

Infinite products

1. John Wallis' product
   2⋅2⋅4⋅4⋅6⋅6⋅…1⋅3⋅3⋅5⋅5⋅7⋅…=π2.
2. François Viète's product
   12⋅2√2⋅2+2√√2⋅2+2+2√√√2⋅2+2+22√√√√2⋅…=1π.
3. ∏n≥2(1−n−2)
   (1−122)⋅(1−132)⋅(1−142)⋅…=12.
4. ∏n≥3(1−4n−2)
   (1−432)⋅(1−442)⋅(1−452)⋅…=16.

Continued fractions

1. π
   1+122+322+522+722+…=4π.
2. more π
   1+123+225+327+429+…=4π.
3. Golden ratio
   1+11+11+11+…=ϕ=1+5√2.

What not

1. Golden ratio
   1+1+1+1√+…−−−−−−−−−−√−−−−−−−−−−−−−−−√−−−−−−−−−−−−−−−−−−−−√=ϕ=1+5√2.