Fun Math

fun.md

Fun Math

Fun Fact 1

• 1 - 1 + 1 - 1 + 1 - 1 + .... = 1/2
• Proof (1) by Luigi Grandi:

S = 1 - 1 + 1 - 1 + 1 - 1 + ...
1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + ...)
= 1 - 1 + 1 - 1 + 1 - 1 + ...
= S

1 - S = S
S = 1/2

• Proof (2):

Convergent series sum =
1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

sum of 1 to n = 2 - 1/n

if n -> infinity, sum = 2

Another way to calculate convergent series sum = partial sums method =

1, (1/2) * (1 + 3/2), (1/3) * (1 + 3/2 + 7/4) + ....

each term = 2 - (1/n) * \sigma_1^n (1/i)

Use this for Grandi's series:

1, (1 + 0)/2, (1 + 0 + 1)/3, (1 + 0 + 1 + 0)/4, ... ----> 1/2

Fun Fact 2

• 1 + 2 + 3 + 4 + ... = -1/12

• Used often in string theory

• Proof:

S_1 = 1 - 1 + 1 - 1 + ... ( = 1/2)
S_2 = 1 - 2 + 3 - 4 + 5 - 6 + ....
S = 1 + 2 + 3 + 4 + 5 + 6 + ....

S_2 + S_2
= 1 - 2 + 3 - 4 + 5 - 6 - ... +
1 - 2 + 3 - 4 + 5 + ...
= 1 - 1 + 1 - 1 + 1 - 1 + ..

2*S_2 = 1/2
S_2 = 1/4

S - S_2 = 1 + 2 + 3 + 4 + 5 + 6 + ...
- (1 - 2 + 3 - 4 + 5 - 6 + ....)
= 4 + 8 + 12 ....
= 4*S

S - S_2 = 4S
3S = -S_2 = -1/4
S = -1/12